optimization problems worksheet

In the previous examples, we considered functions on closed, bounded domains. Example $\PageIndex{5}$: Minimizing Travel Time. Solution We want to build a box whose base length is 6 times the base width and the box will enclose 20 in 3. Great to use as demonstrations using a projector and computer or even with an Interactive Whiteboard. Introduce all variables. Clearly $A(0)=0$ and $A(50)=0$, whereas $A(25) = 625 \text{ft}^2$. An open-topped cylindrical can is to have volume $300\, cm^3$. Step 5: From Figure $\PageIndex{5}$, we see that $0x6$. When $x=6\sqrt[3]{2}$, $y=\dfrac{216}{(6\sqrt[3]{2})^2}=3\sqrt[3]{2}in$. Step 6: As mentioned earlier, $A(x)$ is a continuous function over the closed, bounded interval $[0,2]$. Solving the constraint equation for $y$, we have $y=\dfrac{216}{x^2}$. We motivated our interest in such values by discussing how it made sense to want to know the highest/lowest values of a stock or the fastest/slowest an object was moving. x, x For example, companies often want to minimize production costs or maximize revenue. We will follow the steps outlined by Key Idea 6. The basic idea of the optimization problems that follow is the same. Therefore, the dimensions of the box should be x=623in.x=623in. Since $x=66/\sqrt{55}$ does satisfy that equation, we conclude that $x=66/\sqrt{55}$ is a critical point, and it is the only one. We conclude that the domain is the open, unbounded interval $(0,)$. We start with a classic example which is followed by a discussion of the topic of optimization. Both analytical and graphical approaches are used to study the rate of change. Draw a picture, label variables and write down a constrained optimization problem that models this problem. Find the profit function for the number of pizzas. Example $\PageIndex{3}$: Optimization: minimizing cost. Note that as xx becomes large, the height of the box yy becomes correspondingly small so that x2y=216.x2y=216. Therefore, $C(r)=0$ when $2r=\dfrac{3000}{r^2}$. Step 2: We need to minimize the surface area. Optimization Worksheets. These extreme values occur either at endpoints or critical points. Evaluating $c(x)$ at $x=416.67$ gives a cost of about $370,000. Perfect supplement for any Calculus curriculum, 8 of the 10 word, include the diagram for students to label and to help them solidify their understanding. Step 1: Draw a rectangular box and introduce the variable $x$ to represent the length of each side of the square base; let $y$ represent the height of the box. Let $T$ be the time it takes to get from the cabin to the island. For example, if youarending the smallest surface areaS, then you want to nd an equation forSas a function of onevariable. Since $x=66/\sqrt{55}$ does satisfy that equation, we conclude that $x=66/\sqrt{55}$ is a critical point, and it is the only one. Modify the area function $A$ if the rectangle is to be inscribed in the unit circle $x^2+y^2=1$. Your instructor might use some of these in class. For the following exercises, consider a pizzeria that sell pizzas for a revenue of R(x)=axR(x)=ax and costs C(x)=b+cx+dx2,C(x)=b+cx+dx2, where xx represents the number of pizzas. Let $V$ be the volume of the resulting box. As you increase rent by $25/month,$25/month, one fewer apartment is rented. If the absolute maximum occurs at an interior point, then we have found an absolute maximum in the open interval. To construct a rectangular garden, we certainly need the lengths of both sides to be positive. that uses maximum and minimum skills. Note that as $x$ becomes large, the height of the box $y$ becomes correspondingly small so that $x^2y=216$. A rectangle is to be inscribed in the ellipse. }\) With these dimensions, the surface area is, \[S(6\sqrt[3]{2})=\dfrac{864}{6\sqrt[3]{2}}+(6\sqrt[3]{2})^2=108\sqrt[3]{4}\,\text{in}^2\nonumber \], Consider the same open-top box, which is to have volume $216\,\text{in}^3$. How far should you run west to minimize the time needed to reach the island? x consent of Rice University. What is the minimum surface area? Lecture Notes Optimization 1 page 4 Sample Problems - Solutions 1. Two of the, involves outside work gathering data. What are the dimensions of the rectangular pen to minimize the amount of material needed? You have 400ft400ft of fencing to construct a rectangular pen for cattle. Why do you need to check the endpoints for optimization problems? 9) A closed rectangular container with a square base is to have a volume of 300 in3. Taking the derivative of $A(x)$, we obtain, \[ \begin{align*} A'(x) &=2\sqrt{4x^2}+2x\dfrac{1}{2\sqrt{4x^2}}(2x) \\[4pt] &=2\sqrt{4x^2}\dfrac{2x^2}{\sqrt{4x^2}} \\[4pt] &=\dfrac{84x^2}{\sqrt{4x^2}} . You may use the provided box to sketch the problem setup and theprovided graph to sketch the function of one variable to be minimized or maximized. Determine the height of the box that will give a maximum volume. ; Draw a picture (as always when working with word problems); Identify what is known and unknown, and assign variables to the unknown quantities. If the cost of one of the sides is $30/\text{in}^2,$ the cost of that side is $0.30xy$ dollars. A wire of length 12 inches can be bent into a circle, a square, or cut to make both a circle and a square. How many pizzas sold maximizes the profit? Worksheets. We want the area to be nonnegative. Use rr to represent the radius of the semicircle. Since we earlier found $y = 50-x$, we find that $y$ is also $25$. At what speed is fuel consumption minimized? Therefore, we need $x>0$. Solve the resulting problem using the method of28. The derivative is $R(p)=10p+1000.$ Therefore, the critical point is $p=100$. What should the dimensions of the rectangle be to maximize its area? We do not yet know how to handle functions with 2 variables; we need to reduce this down to a single variable. However, we also have some auxiliary condition that needs to be satisfied. But it models well the necessary process: create equations that describe a situation, reduce an equation to a single variable, then find the needed extreme value. \label{ex3eq2} \], Squaring both sides of this equation, we see that if $x$ satisfies this equation, then $x$ must satisfy, We conclude that if $x$ is a critical point, then $x$ satisfies, [Note that since we are squaring, $ (x-6)^2 = (6-x)^2.$], Therefore, the possibilities for critical points are, Since $x=6+6/\sqrt{55}$ is not in the domain, it is not a possibility for a critical point. A particle is traveling along the -axis and it's position from the origin can be modeled by : ; L F 6 7 7 6 E 121 where is meters and is minutes on the interval . Some of them can also, , are difficult for students. Step 2: We need to minimize the surface area. Find the positive integer that minimizes the sum of the number and its reciprocal. }\) by $30\,\text{in. Step 5: From Figure \(\PageIndex{7}$, we see that to inscribe a rectangle in the ellipse, the $x$-coordinate of the corner in the first quadrant must satisfy $00$, we have $y=\sqrt{\dfrac{1x^2}{4}}$. A patients pulse measures 70 bpm, 80 bpm, then 120 bpm.70 bpm, 80 bpm, then 120 bpm. Label variables and indicate eventual constants. Therefore, the total time spent traveling is, Step 4: From Figure $\PageIndex{5}$, the line segment of $y$ miles forms the hypotenuse of a right triangle with legs of length $2$ mi and $6x$ mi. What are the dimensions of thepen built this way that has the largest area? Therefore, the absolute maximum occurs at $p=$100$. Step 3: As mentioned in step $2$, are trying to maximize the volume of a box. CALCULUS WORKSHEET ON OPTIMIZATION Work the following on notebook paper. Write any equations relating the independent variables in the formula from step, Identify the domain of consideration for the function in step, Locate the maximum or minimum value of the function from step. are not subject to the Creative Commons license and may not be reproduced without the prior and express written Therefore, we consider $V$ over the closed interval $[0,12]$ and check whether the absolute maximum occurs at an interior point. then you must include on every digital page view the following attribution: Use the information below to generate a citation. Boundary Value Problems & Fourier Series, 8.3 Periodic Functions & Orthogonal Functions, 9.6 Heat Equation with Non-Zero Temperature Boundaries, 1.14 Absolute Value Equations and Inequalities. Creative Commons Attribution-NonCommercial-ShareAlike License Step 1: Let $x$ be the distance running and let $y$ be the distance swimming (Figure $\PageIndex{5}$). [T] Where is the parabola y=x2y=x2 closest to point (2,0)?(2,0)? For the following exercises, set up and evaluate each optimization problem. We want to build a box whose base length is 6 times the base width and the box will enclose 20 in, We want to construct a cylindrical can with a bottom but no top that will have a volume of 30 cm. We'll call this the. Find at what angle the lifeguard should swim to reach the drowning person in the least amount of time. How many smartphones Therefore, the absolute maximum occurs at $p=$100$. are licensed under a, Derivatives of Exponential and Logarithmic Functions, Integration Formulas and the Net Change Theorem, Integrals Involving Exponential and Logarithmic Functions, Integrals Resulting in Inverse Trigonometric Functions, Volumes of Revolution: Cylindrical Shells, Integrals, Exponential Functions, and Logarithms. }\) The maximum volume is, \[V(102\sqrt{7})=640+448\sqrt{7}1825\,\text{in}^3. Objective: Create a function and find its maximum value. Find the dimensions of the box that minimize cost. The car rental company should charge $$100$ per day per car to maximize revenue as shown in the following figure. To download/print, click on pop-out icon or print icon to worksheet to print or download. Except where otherwise noted, textbooks on this site We earlier set $y = 50-x/2$; thus $y = 25$. Therefore, the cost to produce the can is. by $36$ in. The area of each of the four vertical sides is $xy.$ The area of the base is $x^2$. Pre-made digital activities. Therefore, we conclude that $T$ has a local minimum at $x5.19 mi$. Find the volume of the largest right circular cylinder that fits in a sphere of radius 1.1. It has found that the number of cars rented per day can be modeled by the linear function $n(p)=7505p.$ How much should the company charge each customer to maximize revenue? $C=(5) 2 \pi rh+ (10)\pi r^2$, with $r \geq 0$. This minimum must occur at a critical point of $S$. Eschew memorizing how to do "this kind of problem" as opposed to "that kind of problem." Therefore, it has an absolute maximum (and absolute minimum). This an activity that uses math skills in real world context. Therefore, we need $r>0$. Step 5: Since the owners plan to charge between $$50$ per car per day and $$200$ per car per day, the problem is to find the maximum revenue $R(p)$ for $p$ in the closed interval $[50,200]$. This content by OpenStax is licensedwith a CC-BY-SA-NC4.0license. Step 6: Since $V(x)$ is a continuous function over the closed, bounded interval $[0,12], V$ must have an absolute maximum (and an absolute minimum). Expect to see the farmer problem and the open-top box problem To advance in the circuit, students must find their answer, and with that answer is a new problem. The file contains the following: (2,excel files, 7 geogebra applets and 7 html files that can be viewed in Internet Explorer, Firefox or Chrome) The descriptions of these files are all found below: The area of each of the four vertical sides is $xy.$ The area of the base is $x^2$. This book uses the Optimization problems tend to pack loads of information into a short problem. We can now write Area as \[\text{Area} = A(x) = x(50-x/2) = 50x - \frac12x^2.\[ Area is now defined as a function of one variable. OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. For the following exercises, consider a limousine that gets m(v)=(1202v)5mi/galm(v)=(1202v)5mi/gal at speed v,v, the chauffeur costs $15/h,$15/h, and gas is $3.5/gal.$3.5/gal. ; Determine what value needs to be optimized (maximized or minimized). The Objective Equation is the equation that illustrates the object of the problem. The anchor charts provide students with steps for solving for absolute extrema by checking endpoints, and when needed, using the second derivative test.The no prep lesson and worksheet are easy for students to follow and complete. The volume of a box is. y The eventual goal is to arrive atafunction of one variablerepresenting a quantity to be optimized. We know more about the situation: the man has 100 feet of fencing. If applicable, draw a figure and label all variables. CENTERS & WORKSHEETS. You have a garden row of 2020 watermelon plants that produce an average of 3030 watermelons apiece. It is not difficult to show that for a closed-top box, by symmetry, among all boxes with a specified volume, a cube will have the smallest surface area. Express the variable to be optimized as a function of the variables you used in part (b). It has found that the number of cars rented per day can be modeled by the linear function $n(p)=7505p.$ How much should the company charge each customer to maximize revenue? Write a function for each problem, and justify your answers. In this section, we show how to set up these types of minimization and maximization problems and solve them by using the tools developed in this chapter. }\); otherwise, one of the flaps would be completely cut off. Find the dimensions of a right cone with surface area S=4S=4 that has the largest volume. Therefore, we need to minimize $S$. Since $x=\sqrt{2}$ is a solution of Equation \ref{ex5eq1}, we conclude that $\sqrt{2}$ is the only critical point of $A(x)$ in the interval $[0,2]$. This circuit has 12 word, which start easy and build from there. We conclude that the domain is the open, unbounded interval (0,).(0,). The distance the power line is laid along land is $5000-416.67 = 4583.33$ ft., and the underwater distance is $\sqrt{416.67^2+1000^2} \approx 1083$ ft. We now find the extreme values. Locate the maximum or minimum value of the function from step $4.$ This step typically involves looking for critical points and evaluating a function at endpoints. When x=623,x=623, y=216(623)2=323in.y=216(623)2=323in. In the exercises, you will see a variety of situations that require you to combine problem--solving skills with calculus. The derivative is. Therefore, $x^2=2.$ Thus, $x=\sqrt{2}$ are the possible solutions of Equation. This minimum must occur at a critical point of S.S. Clearly identify what quantity is to be maximized or minimized. square feet Stuck? Certainly, we need $x>0.$ Furthermore, the side length of the square cannot be greater than or equal to half the length of the shorter side, $24\,\text{in. To find the maximum value, look for critical points. Therefore, \(A(x)$ must have an absolute maximum at the critical point $x=\sqrt{2}$. Quizzes with auto-grading, and real-time student data. y. to run a power line along the land, and $130/ft. We are going to fence in a rectangular field. Solving optimization problems Optimization AP.CALC: FUN4 (EU), FUN4.B (LO), FUN4.B.1 (EK), FUN4.C (LO), FUN4.C.1 (EK) Google Classroom An open-topped glass aquarium with a square base is designed to hold 62.5 62.5 cubic feet of water. The maximum area is $5000\, \text{ft}^2$. where $L,\,W,$and $H$ are the length, width, and height, respectively. How much of the power line should be run along the land to minimize the overall cost? Taking the derivative of $A(x)$, we obtain, \[ \begin{align*} A'(x) &=2\sqrt{4x^2}+2x\dfrac{1}{2\sqrt{4x^2}}(2x) \\[5pt] &=2\sqrt{4x^2}\dfrac{2x^2}{\sqrt{4x^2}} \\[5pt] &=\dfrac{84x^2}{\sqrt{4x^2}} . However, we do know that a continuous function has an absolute maximum (and absolute minimum) over a closed interval. See the link for each individual month to see an expanded preview for that month.These, review topics that are traditionally taught during Calculus AB. *Click on Open button to open and print to worksheet. By choosing $x$ as we did, we make the expression under the square root simple. You will find many other activity sheets for calculus available in my store. Identify the values of all relevant quantities of the problem. Therefore, we need $x>0$. However, in the next step, we discover why this function must have an absolute minimum over the interval (0,).(0,). They will be optimizing a soda can, rectangular-base cardboard box, and fencing in a farm. For non-calculus students a grapher will be necessary. We have a piece of cardboard that is 50 cm by 20 cm and we are going to cut out the corners and fold up the sides to form a box. This formula may involve more than one variable. The plush material for the square bottom of the box costs $5/ft2$5/ft2 and the material for the sides costs $2/ft2.$2/ft2. Our enclosure is sketched twice in Figure $\PageIndex{1}$, either with green grass and nice fence boards or as a simple rectangle. Solution A piece of pipe is being carried down a hallway that is 18 feet wide. Find the dimensions of the closed cylinder volume V=16V=16 that has the least amount of surface area. In many questions, students are required to communicate their understanding of the theorems verbally. 2. Find the dimensions of the rectangular field of largest area that can be fenced. If asked to minimize cost, an In the perimeter equation, solve for $y$: $y = 50 - x/2$. Step 1: Draw a cylindrical can and introduce the variable $r$ to represent the radius of the circular base; let $h$ represent the height of the can. Suppose the cost of the material for the base is $20/in.^2$ and the cost of the material for the sides is $30/in.^2$ and we are trying to minimize the cost of this box. Therefore, the surface area of the box is, Step 4: Since the volume of this box is x2yx2y and the volume is given as 216in.3,216in.3, the constraint equation is. A visitor is staying at a cabin on the shore that is $6$ mi west of that point. Lets use these data to determine the price the company should charge to maximize the amount of money it brings in. Therefore, we check whether $\sqrt{2}$ is a solution of Equation. Understanding the principles here will provide a good foundation for the mathematics you will likely encounter later. Therefore, the only critical point is $x=25$ (Figure $\PageIndex{2}$). For the following problems, consider a lifeguard at a circular pool with diameter 40m.40m. Since Distance = Rate Time $(D=RT),$ the time spent running is. The latter inequality implies that $x\leq100$, so $0\leq x\leq 100$. We will see that, although the domain of consideration is $(0,),$ the function has an absolute minimum. (limits, finding a derivative, chain rule, derivatives of trig functions, position, velocity, acceleration, related rates, op, Give your students engaging practice with the circuit format! True or False. Therefore, the dimensions of the box should be $x=6\sqrt[3]{2}in.$ and $y=3\sqrt[3]{2}in.$ With these dimensions, the surface area is, \[S(6\sqrt[3]{2})=\dfrac{864}{6\sqrt[3]{2}}+(6\sqrt[3]{2})^2=108\sqrt[3]{4}in.^2\], Consider the same open-top box, which is to have volume $216in.^3$. This has the undesired effect of having the longest distance of all, probably ensuring a non--minimal cost. Therefore, we can write the cost as a function of \ (r\) only: $C=(5) 2 \pi r\dfrac{300}{\pi r^2}+ (10)\pi r^2$. One common application of calculus is calculating the minimum or maximum value of a function. When $p=200, R(p)=$0$. Write the cost as a function of the side lengths of the base. $T_{swimming}=\dfrac{D_{swimming}}{R_{swimming}}=\dfrac{y}{3}$. In business, companies are interested in maximizing revenue. With these dimensions, the surface area is, Consider the same open-top box, which is to have volume 216in.3.216in.3. Example $\PageIndex{7}$: Maximizing the Area of an Inscribed Rectangle, A rectangle is to be inscribed in the ellipse. Since $S$ is a continuous function that approaches infinity at the ends, it must have an absolute minimum at some $x(0,)$. 1) A company has started selling a new type of smartphone at the price of $ 110 0.05 x where x is the number of smartphones manufactured per day. Therefore, we consider the following problem: Maximize $A(x)=100x2x^2$ over the interval $[0,50].$, As mentioned earlier, since $A$ is a continuous function on a closed, bounded interval, by the extreme value theorem, it has a maximum and a minimum. Example $\PageIndex{1}$: Optimization: perimeter and area. Therefore, we consider $V$ over the closed interval $[0,12]$ and check whether the absolute maximum occurs at an interior point. Beginning Calculus students can really struggle with. (Let $x$ be the side length of the base and $y$ be the height of the box.). What are the dimensions of the box with the largest volume? An island is $2\,mi$ due north of its closest point along a straight shoreline. Either way, drawing a rectangle forces us to realize that we need to know the dimensions of this rectangle so we can create an area function -- after all, we are trying to maximize the area. On the other hand, $x$ is allowed to have any positive value. In the following example, we look at constructing a box of least surface area with a prescribed volume. Find a function of one variable to describe the quantity that is to be minimized or maximized. This minimum must occur at a critical point of $C$. The following is a list of worksheets and other materials related to Math 122B and 125 at the UA. We need to label our unknown distances -- the distance run along the ground and the distance run underwater. Give all decimal answers correct to three decimal places. My students love this format! Suppose the visitor runs at a rate of $8$ mph and swims at a rate of $3$ mph. How many pizzas sold maximizes the profit? OPTIMIZATION - PART 1 NAME_____ 1. Step 2: The problem is to maximize $R.$, Step 3: The revenue (per day) is equal to the number of cars rented per day times the price charged per car per daythat is, $R=np.$, Step 4: Since the number of cars rented per day is modeled by the linear function $n(p)=10005p,$ the revenue $R$ can be represented by the function, \[ \begin{align*} R(p) &=np \\[5pt] &=(10005p)p \\[5pt] &=5p^2+1000p.\end{align*}\]. Write the cost as a function of the side lengths of the base. \[T(x)=\dfrac{x}{6}+\dfrac{\sqrt{(15x)^2+1}}{2.5} \nonumber \]. When you find the maximum for an optimization problem, why do you need to check the sign of the derivative around the critical points? Choose xx to minimize the sum of their areas. Two examples illustrate step by step procedures that can be followed to solve most, . You are constructing a box for your cat to sleep in. Many functions still have at least one absolute extrema, even if the domain is not closed or the domain is unbounded. Many functions still have at least one absolute extrema, even if the domain is not closed or the domain is unbounded. If the fundamental equation defines the quantity to be optimized as a function of more than one variable, reduce it to a single variable function using substitutions derived from the other equations. This formula may involve more than one variable. $T_{swimming}=\dfrac{D_{swimming}}{R_{swimming}}=\dfrac{y}{3}$. The optimal solution likely has the line being run along the ground for a while, then underwater, as the figure implies. Similarly, as $x$ becomes small, the height of the box becomes correspondingly large. Since Distance = Rate Time $(D=RT),$ the time spent running is. Step 3: The area of the rectangle is $A=LW.$, Step 4: Let $(x,y)$ be the corner of the rectangle that lies in the first quadrant, as shown in Figure $\PageIndex{7}$. R, This activity sheet has 10 conceptually based questions on solving, . What is the longest pipe (always keeping it horizontal) that can be carried around the turn in the hallway? Solution A man has 100 feet of fencing, a large yard, and a small dog. Optimization Problems Practice Solve each optimization problem. Therefore, the volume of the box is, \[ \begin{align*} V(x) &=(362x)(242x)x \\[5pt] &=4x^3120x^2+864x \end{align*}.\]. To maximize revenue, a car rental company has to balance the price of a rental against the number of cars people will rent at that price. Similarly, as xx becomes small, the height of the box becomes correspondingly large. y At the endpoints $x=0$ and $x=2$, $A(x)=0.$ For $00$. citation tool such as, Authors: Gilbert Strang, Edwin Jed Herman. For calculus students techniques of differential calculus are suitable.Problem scenarios include: volumes, cost, surface area. . Recognize that this is never undefined. y Therefore, by the Pythagorean theorem, $2^2+(6x)^2=y^2$, and we obtain $y=\sqrt{(6x)^2+4}$. How much wire should be used for the circle if the total area enclosed by the figure(s) is to be a minimum? Consequently, by the extreme value theorem, we were guaranteed that the functions had absolute extrema. The student will be given a function and will be asked to list the points at which that the tangent line to that function is horizontal. Then, the remaining four flaps can be folded up to form an open-top box. ; otherwise, one of the flaps would be completely cut off. We will proceed to show how calculus can provide this answer in a context that proves this answer is correct. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Step 6: Note that as x0+,x0+, S(x).S(x). In this section, we apply the concepts of extreme values to solve "word problems," i.e., problems stated in terms of situations that require us to create the appropriate mathematical framework in which to solve the problem. Suppose the dimensions of the cardboard in Example $\PageIndex{2}$ are 20 in. What is the minimal cost? Solution Two 10 meter tall poles are 30 meters apart. A window is composed of a semicircle placed on top of a rectangle. Solving this equation for $r$, we obtain $r^3=1500$, so $r=\sqrt[3]{1500}.$ Since this is the only critical point of $C$, the absolute minimum must occur at $r=\sqrt[3]{1500}$. Created by T. Madas Created by T. Madas Question 3 (***) The figure above shows a solid brick, in the shape of a cuboid, measuring 5x cm by x cm by h cm . 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MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FCalculus%2FMap%253A_Calculus__Early_Transcendentals_(Stewart)%2F04%253A_Applications_of_Differentiation%2F4.07%253A_Optimization_Problems, $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}$ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$, Example $\PageIndex{1}$: Maximizing the Area of a Garden, Problem-Solving Strategy: Solving Optimization Problems, Example $\PageIndex{2}$: Maximizing the Volume of a Box, Example $\PageIndex{3}$: Minimizing Travel Time, Example $\PageIndex{4}$: Maximizing Revenue, Example $\PageIndex{5}$: Maximizing the Area of an Inscribed Rectangle, Example $\PageIndex{6}$: Minimizing Surface Area, 4.6: Graphing with Calculus and Calculators, Solving Optimization Problems over a Closed, Bounded Interval, Solving Optimization Problems when the Interval Is Not Closed or Is Unbounded. \begin{array}{ccccc} & & \text{\$50}\times \text{land distance} &+& \text{\$130}\times \text{water distance} \\ You'll do this a lot in Math 124 using calculus . Note that, unlike the previous examples, we cannot reduce our problem to looking for an absolute maximum or absolute minimum over a closed, bounded interval. In the next step, we discover why this function must have an absolute minimum over the interval $(0,).$, Step 6: Note that as $x0+^, C(r).$ Also, as $r, C(r)$. Optimization is the process of making a quantity as large or small as possible. True or False. The material for the bottom of the can costs $10\, cents/cm^2$, for its curved side costs $5 \, cents/cm^2.$ Find the dimensions of the can that minimize the cost of the can. When $p=200, R(p)=$0$. For the following exercises, consider two nonnegative numbers xx and yy such that x+y=10.x+y=10. Second, we could minimize the underwater length by running a wire all 5000 ft. along the beach, directly across from the offshore facility. Write a formula for the quantity to be maximized or minimized in terms of the variables. There are two immediate solutions that we could consider, each of which we will reject through "common sense." They have extremely important applications in economics, engineering, and science. Minimizing f(x)=xAx+xb 2 Rn,orsubjecttolinearoranecon- Virge Cornelius' Mathematical Circuit Training, This lesson introduces students to what are known as ", ." There are 160 questions in this package. Find the dimensions of the can that minimize the surface area. Therefore, lets consider the function $A(x)=100x2x^2$ over the closed interval $[0,50]$. 3.6: Applied Optimization Problems is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts. To maximize the area of the garden, we need to find the maximum value of the function, Problem-Solving Strategy: Solving Optimization Problems. Therefore, we need x>0.x>0. It may be helpful to highlight certain values within the problem. As an Amazon Associate we earn from qualifying purchases. If the maximum value occurs at an interior point, then we have found the value $x$ in the open interval $(0,50)$ that maximizes the area of the garden. It includes the, and an explanation.3. \end{align*}\], To find critical points, we need to find where $A'(x)=0.$ We can see that if $x$ is a solution of, \[\dfrac{84x^2}{\sqrt{4x^2}}=0, \label{ex5eq1} \]. Suppose the cost of the material for the base is $20/\text{in}^2$ and the cost of the material for the sides is $30/\text{in}^2$ and we are trying to minimize the cost of this box. An island is $2$ mi due north of its closest point along a straight shoreline. It also includes verification of the solution using Calculu. When $p=100, R(100)=$50,000.$ When $p=50, R(p)=$37,500$. The company should charge $$75$ per car per day. x Step 2: The problem is to maximize $A$. Let $R$ be the revenue per day. Therefore, $x^2=2.$ Thus, $x=\sqrt{2}$ are the possible solutions of Equation \ref{ex5eq1}. However, in the next step, we discover why this function must have an absolute minimum over the interval $(0,).$, Step 6: Note that as $x0^+,\, S(x).$ Also, as $x, \,S(x)$. If they charge $$200$ per day or more, they will not rent any cars. For any additional watermelon plants planted, the output per watermelon plant drops by one watermelon. Example $\PageIndex{4}$: Maximizing the Volume of a Box. It costs $50/ft. and you must attribute OpenStax. For every continuous nonconstant function on a closed, finite domain, there exists at least one xx that minimizes or maximizes the function. What dimensions provide the maximal area? Step 1: Let $p$ be the price charged per car per day and let $n$ be the number of cars rented per day. Write an equation that expresses the quantity to be op-timized in terms of the other quantities and use any con-straints in the problem to eliminate all but one indepen-dent variable. Step 5: From Figure $\PageIndex{5}$, we see that $0x6$. Suppose a visitor swims at the rate of $2.5\,mph$ and runs at a rate of $6\,mph$. Solving the constraint equation for $y$, we have $y=\dfrac{216}{x^2}$. Gilbert Strang (MIT) and Edwin Jed Herman (Harvey Mudd) with many contributing authors. An open-top box is to be made from a $24\,\text{in. Use these equations to write the quantity to be maximized or minimized as a function of one variable. Therefore, the problem reduces to looking for the maximum value of \(A(x)$ over the open interval $(0,2)$. To justify that the time is minimized for this value of $x$, we just need to check the values of $T(x)$ at the endpoints $x=0$ and $x=6$, and compare them with the value of $T(x)$ at the critical point $x=66/\sqrt{55}$. Add highlights, virtual manipulatives, and more. The cost of the material of the sides is $3/in 2 and the cost of the top and bottom is $15/in 2. Worksheets are Calc, Work 24 optimization, Optimization date period, Name panther id optimization work calculus i, Problems and solutions in optimization, Word problems with maxmin, Calculus 1 optimization problems, A generic function. Step 5: Since we are requiring that $\pi r^2 h=300$, we cannot have $r=0$. Look for critical points to locate local extrema. A rectangular box with a square base, an open top, and a volume of $216 in.^3$ is to be constructed. For example, companies often want to minimize production costs or maximize revenue. Let $S$ denote the surface area of the open-top box. They are intended to be done in partners and/or small groups. To find the maximum value, look for critical points. On the other hand, xx is allowed to have any positive value. Step 6: Since $T(x)$ is a continuous function over a closed, bounded interval, it has a maximum and a minimum. Solving the constraint equation for $h$, we have $h=\dfrac{300}{\pi r^2}$. Step 6: Since $V(x)$ is a continuous function over the closed, bounded interval $[0,12]$, $V$ must have an absolute maximum (and an absolute minimum). Solution One can likely guess the correct answer -- that is great. http://www.apexcalculus.com/, Pamini Thangarajah(MountRoyal University, Calgary, Alberta, Canada). Legal. (One of these should describe the quantity to be optimized. a. Step 6: Since $R$ is a continuous function over the closed, bounded interval $[50,200]$, it has an absolute maximum (and an absolute minimum) in that interval. Therefore, the area is, $A=LW=(2x)(2y)=4x\sqrt{\dfrac{1x^2}{4}}=2x\sqrt{4x^2}$. This content iscopyrighted by a Creative CommonsAttribution - Noncommercial (BY-NC) License. We will proceed to show how calculus can provide this answer in a context that proves this answer is correct. Set up and solve optimization problems in several applied fields. 1999-2023, Rice University. If you have a river on one side of your property, what is the dimension of the rectangular pen that maximizes the area? Let $A$ be the area of the rectangle. Topics covered: Each month contains 20 questions. The area of this rectangle is$ A=LW=(2\sqrt{2})(\sqrt{2})=4.$. We evaluate $A(x)$ at the endpoints of our interval and at this critical point to find the extreme values; in this case, all we care about is the maximum. 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Then we have $y=1002x=1002(25)=50.$ To maximize the area of the garden, let $x=25\,\text{ft}$ and $y=50\,\text{ft}$. Lets now consider functions for which the domain is neither closed nor bounded. Therefore, we are trying to determine whether there is a maximum volume of the box for $x$ over the open interval $(0,12).$ Since $V$ is a continuous function over the closed interval $[0,12]$, we know $V$ will have an absolute maximum over the closed interval. Assume that R(x)=15x,R(x)=15x, and C(x)=60+3x+12x2.C(x)=60+3x+12x2. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Therefore, we are trying to determine whether there is a maximum volume of the box for $x$ over the open interval $(0,12).$ Since $V$ is a continuous function over the closed interval $[0,12]$, we know $V$ will have an absolute maximum over the closed interval. h(t) is quadratic, with a negative leading coe . $R(p)=np,$ where $n$ is the number of cars rented and $p$ is the price charged per car. What is the maximum area? If asked to maximize area, an equation representing the total area is your objective equation. On the other hand, $x$ is allowed to have any positive value. Our enclosure is sketched twice in Figure 3.6.1, either with green grass and nice fence boards or as a simple rectangle. Let $C$ denote the cost to produce a can. Suppose the island is $1$ mi from shore, and the distance from the cabin to the point on the shore closest to the island is $15$ mi. A truck uses gas as g(v)=av+bv,g(v)=av+bv, where vv represents the speed of the truck and gg represents the gallons of fuel per mile. Make a sketch if helpful. Therefore, the volume of the box is, \[ \begin{align*} V(x) &=(362x)(242x)x \\[4pt] &=4x^3120x^2+864x \end{align*}. Given $y=f(x)$, they offer a method of approximating the change in $y$ after $x$ changes by a small amount. x Included is a step-by-step problem that utilizes the 8 steps. ------------------------------------------------------------------------------------------------ Example: Optimization 1 rancher wants to build a rectangular pen, using one side of her barn for one side of thepen, and using 100m of fencing for the other three sides. We conclude that the maximum area must occur when $x=25$. y Write a formula for the quantity to be maximized or minimized in terms of the variables. y. Therefore, $[0,6]$ is the domain of consideration. I encouraged my students to do every one of their, on this worksheet so that they would have the steps clearly in their mind. Then, the remaining four flaps can be folded up to form an open-top box. Therefore, $S(x)=0$ when $2x=\dfrac{864}{x^2}$. This a great activity that can be done in small groups. }\) piece of cardboard by removing a square from each corner of the box and folding up the flaps on each side. Therefore, the volume is maximized if we let $x=102\sqrt{7}\, in.$ The maximum volume is, \[V(102\sqrt{7})=640+448\sqrt{7}1825\,in.^3 \nonumber \]. Step 3: To find the time spent traveling from the cabin to the island, add the time spent running and the time spent swimming. When giving an exam, I did not provide th. Calculus Optimization Problems/Related Rates Problems Solutions 1) A farmer has 400 yards of fencing and wishes to fence three sides of a rectangular field (the fourth side is along an existing stone wall, and needs no additional fencing). To solve an optimization problem, begin by drawing a picture and introducing variables. Sketch a diagram and label relevant quantities. We can write length $L=2x$ and width $W=2y$. Owners of a car rental company have determined that if they charge customers $p$ dollars per day to rent a car, where $50p200$, the number of cars $n$ they rent per day can be modeled by the linear function $n(p)=10005p$. It helps to make a sketch of the situation. Therefore, we are trying to determine the maximum value of $A(x)$ for $x$ over the open interval $(0,50)$. Creative Commons Attribution-NonCommercial-ShareAlike License, https://openstax.org/books/calculus-volume-1/pages/1-introduction, https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems, Creative Commons Attribution 4.0 International License. Identify the domain of this function, keeping in mind the context of the problem. Where should the wire be anchored to the ground to minimize the amount of wire needed? The surface area of the curved side is $2 \pi rh$ The area of the bottom is $\pi r^2$. Suppose the dimensions of the cardboard in Example $\PageIndex{2}$ are $20\,\text{in. Pre-Calculus Optimization Problems Fencing Problems A farmer has 480 meters of fencing with which to build two animal pens with a common side as shown in the diagram. Optimization Name___________________________________ Date________________ Period____ Solve each optimization problem. In addition to my Free Hydro Project, this package includes 5 more great real life, for calculus students to solve. We conclude that the domain is the open, unbounded interval \((0,)$. A rectangular box with a square base, an open top, and a volume of $216 \,\text{in}^3$ is to be constructed. Find two positive numbers whose product is 750 and for which the sum of one and 10 times the other is a minimum. Therefore, $S(x)=0$ when $2x=\dfrac{864}{x^2}$. It is not difficult to show that for a closed-top box, by symmetry, among all boxes with a specified volume, a cube will have the smallest surface area. On the other hand, $r$ is allowed to have any positive value. Step 1: Let $x$ be the side length of the square to be removed from each corner (Figure $\PageIndex{3}$). Legal. To determine an accurate measurement of pulse, the doctor wants to know what value minimizes the expression (x70)2+(x80)2+(x120)2?(x70)2+(x80)2+(x120)2? & & 50(5000-x) &+& 130\sqrt{x^2+1000^2}.\\ By the end of this lesson you will be able to: use the first and second derivative to optimize practical, of the DayThis is a bundle of all of my Calculus, of the Day. Step 3: The area of the rectangle is $A=LW.$, Step 4: Let $(x,y)$ be the corner of the rectangle that lies in the first quadrant, as shown in Figure $\PageIndex{7}$. What dimensions provide the maximal area? Owners of a car rental company have determined that if they charge customers $p$ dollars per day to rent a car, where $50p200$, the number of cars $n$ they rent per day can be modeled by the linear function $n(p)=10005p$. A maximum? We can use a graph to determine the dimensions of a box of given the volume and the minimum surface area. On the other hand, $x=66/\sqrt{55}$ is in the domain. Now lets look at a general strategy for solving optimization problems similar to Example $\PageIndex{1}$. Worksheets are Optimization date period, Calc, Math 136 optimization problems exercises, Math 1a calculus work, Calculus optimization work, Work on optimization and related rates, Math 102 chapter optimization word problems, Work on optimization problems. Step 6: As mentioned earlier, $A(x)$ is a continuous function over the closed, bounded interval $[0,2]$. Quadratic Optimization Problems 12.1 Quadratic Optimization: The Positive DeniteCase In this chapter, we consider two classes of quadratic opti-mization problems that appear frequently in engineeringand in computer science (especially in computer vision): Minimizing over all x straints. Since $A(x)$ will have an absolute maximum (and absolute minimum) over the closed interval $[0,2]$, we consider $A(x)=2x\sqrt{4x^2}$ over the interval $[0,2]$. $T_{running}=\dfrac{D_{running}}{R_{running}}=\dfrac{x}{8}$. We need to minimize the cost. Let $x$ denote the length of the side of the garden perpendicular to the rock wall and $y$ denote the length of the side parallel to the rock wall. Lets now consider functions for which the domain is neither closed nor bounded. For example, in Example $\PageIndex{1}$, we are interested in maximizing the area of a rectangular garden. Let $R$ be the revenue per day. Other needs: Starbursts, scissors, rulers, tape What size square should be cut out of each corner to get a box with the maximum volume? Download for free at http://cnx.org. Add answer text here and it will automatically be hidden if you have a "AutoNum" template active on the page. If they charge $$200$ per day or more, they will not rent any cars. Differentiating the function $A(x)$, we obtain. from an Algebra 2/Precalculus perspective, from the perspective of Algebra 2 and Precalculus with Calculus verification. Understand the problem. Are you getting the free resources, updates, and special offers we send out every week in our teacher newsletter? \nonumber \]. We want to maximize the area; as in the example before, \[\text{Area} = xy.\[ This is our fundamental equation. Therefore, we can still consider functions over unbounded domains or open intervals and determine whether they have any absolute extrema. We find that $T(0)2.108h$ and $T(6)1.417h$, whereas. Therefore, the dimensions of the can sho uld be $r=\sqrt[3]{1500}in.$ and $ h= \dfrac{300}{\pi {\sqrt[3]{1500}}^2}$. When $x=6\sqrt[3]{2}$, $y=\dfrac{216}{(6\sqrt[3]{2})^2}=3\sqrt[3]{2}\,\text{in. 2 Lets use these data to determine the price the company should charge to maximize the amount of money it brings in. Determine your Objective Equation. We now create the cost function. Therefore, we can write the surface area as a function of xx only: Step 5: Since we are requiring that x2y=216,x2y=216, we cannot have x=0.x=0. \[c(0) = 380,000 \quad\quad c(5000) \approx 662,873.\], We now find the critical values of \(c(x)$. Accessibility StatementFor more information contact us atinfo@libretexts.org. The file contains a pdf of all of the questions. The area of the base is x2.x2. BUNDLE. Certainly, if we keep making the side lengths of the garden larger, the area will continue to become larger. The derivative is $R(p)=10p+1000.$ Therefore, the critical point is $p=100$ When $p=100, R(100)=$50,000.$ When $p=50, R(p)=$37,500$. Let $V$ be the volume of the resulting box. Therefore, the surface area of the box is, Step 4: Since the volume of this box is $x^2y$ and the volume is given as $216\,\text{in}^3$, the constraint equation is. Consequently, we consider the modified problem of determining which open-topped box with a specified volume has the smallest surface area.
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