2.3 Maximum and Minimum Applications

Transcription

1 Section. 55. Maimum and Minimum Applications Maimizing (or minimizing) is an important technique used in various fields of study. In business, it is important to know how to find the maimum profit and how to minimize loss. In physics, it is important to find values such as the maimum speed of an object, or the minimum distance between two moving bodies. In Calculus, maimization is done by analyzing derivatives of functions, a topic beyond our reach now. However, we can look at these problems numerically, using graphing calculators and the properties of the graph of an equation to find the maimum or minimum values. This section will deal with several different types of these problems. We start off with ma/min problems of a type of that should be familiar to you. These are the ones involving a quadratic equation: y a b c. Eample (Quadratic equations) The height above ground of an object at time t (in seconds) traveling vertically and subject only to the pull of gravity is given by the equation h 6 t v o t h o, where h o is the initial height (in feet) and vo is the initial velocity (in feet per second). A bullet fired upward from ground level has an initial velocity of 480 feet per second. How long does it take to reach its maimum height? How high does it go? Solution: Here are two ways to solve this problem. h 6 t 480 t v o 480, h 0 is quadratic, we o I. Since the equation know the graph of the equation has the following properties: A. The points on the graph are the ordered pairs (t, h), where h is the height at the time t. B. The graph is a parabola which will have a verte. C. Since a = 6 is negative, the graph opens downward. D. Since B and C are true, the verte must be the highest point on the graph (maimum). Both questions will be answered if we find the ordered pair for the verte. This can be done by completing the square or by using the fact that t at the verte. From this formula we find t 5. This ( 6) tells us that it takes 5 seconds for the bullet to reach its maimum height. The maimum height is then found by finding h when t is 5. So h 6(5) 480(5) 600. The maimum height is 600 feet. b a

2 On CD: Graphing Polynomials On CD: Finding the Ma and Min of Polynomials 56 II. We can also solve this by graphing the equation h 6 t 480 t and finding the maimum using a calculator. Here are the steps to accomplish this task. A. Graph Y on the graphing calculator. Remember that X is t and Y is h. B. We must adjust the viewing window to see all the important features of the graph. C. Setting the window as below we get the following graph. D. To find the maimum (which is the verte), we use the maimum finder and get = 5 and y = 600. (Note: When you use a calculator to solve a problem you are introducing round-off errors. The resulting answers are usually not eact.) It takes 5 seconds for the bullet to reach a maimum height of 600 feet. Now we will look at some eamples that deal with cubic and other equations. Note that at this time, we can solve these problems only by using the graphing calculator. Algebraic approaches are introduced in Calculus. Section. 57 Eample (Maimizing volume) An open-top bo is to be made from a sheet of cardboard that measures 0 inches by 5 inches by cutting squares of equal size from each of the corners and then folding the flaps up. What size squares should be removed to maimize the volume of the bo? Solution: 5 0 W L T h is pro blem involves finding the volume of a bo so we need to use the fo rmula for the volume: V = LWH. A. What are the length (L), width (W), and height (H) for our bo? Well, the length of the original sheet of cardboard is 5 inches. In order to make the bo, we will be removing a square of length from each corner and folding along the dotted lines. After doing so, we have that L = 5 = 5. The same process takes place for the width so we have W = 0. The height of the bo is determined by the dimensions of the squares that were removed so H =. Then V ( 5 )(0 ). B. The volume must now be maimized. To do this, we graph the equation in an appropriate window. Note that Y is the volume and is the side length of the squares. (See comments below.)

3 58 C. Use the maimum finder to find the maimum volume for t he bo. When using the maimum finder, we get that the maimum volume is approimately , which is the y-value. The size of the squares that are removed is approimately 4.098, which is the -value. Since we were asked to find the size of the squares, we have th they are inches by inches. at [Co mments:. How did we know to let go from 0 to 0 in the vie wing window? In orde r to construct a bo in this fashion we m re re than 0 inches then we s for Ymin and Yma e, or trace along the graph. u th a ust remove less than 0 inches of material from each corner. If we moved mo wouldn t have a bo! Finding sable value will then require that you look at e tabl. The equa tion that we have for the volum e is not a quadratic so the lgebraic tec hnique used in eam ple does not apply.] Eample (Maimizing area) Given the equation y 8, find the positive value of for which the triangle points (0, 0), (, 0), and (, y) has maim with vertices at the um area. Solution: (, y) (0,0) (, 0) We need to use the formula fo r the area of a triangle A bh Section. 59 A. What are the base b and the height h for our problem? In this case the base is and the height is y. So the area is A y. Since (, y ) is on the graph of y 8, the fo rmula for the area becomes A 8. B. Now we graph the equation in an appropriate window. We choose the values for to go f rom - to 5 sin ce the actual values for must be between 0 and 8. Note that Y here is the area A. C. Use the calculator to find that the maimum y-value for this graph. Remember, the y-value represents the area. So an -value of approimately.6 will give you a triangle with maimum area of approimately 4.5. [Comments : Even though the original equation was a quadratic, the equation that we found for the area was not quadratic. Since 8 is not quadratic, the algebraic form ula for the verte does not apply.].

4 60 Eample 4 ( Minimizing distance) Find the minim um distance from the graph of y and the point (, ). Solution: We need to find the formula for the distance from the point (, ) to all points on the graph. A. The points on the graph are the ordered-pairs (, y) which satisfy y. Then a typical point on the graph is,., we use the distance formula. This says that B. To find the distance d between the points (, ) and d. This is the quantity we want to minimize. C. Graph the equation in the appropriate window, where Y is the distance d given by the formula above. Then y here represents the distance from the point (, ) to the original graph. D. Use the calculator to find the minimum distance. Notice that this graph has two candidates for the minimum value. We are concerned with finding the absolute minimum distance. After some calculator work, we see that this occurs at the low point on the right side of the picture. Section. 6 So we have that when =.864 (appro.), the minimum distance d is approimately T he point on the original graph which is closest to (, ) is now determined by plugging the -value into th orig inal equation y. The resulting closest point is approimately (.864, 0.884). Check the original graph to see if the answer makes sense. e Here s a more involved eample for you. Multipart problems like these often appear in Calculus courses. Eample 5 (Minimizing time) A certa in lifeguard can run at 5.7 feet per second and can swim at 5.9 feet per second. He is located 0 feet away from the edge of the ocean. A swimmer is drowning 5 0 feet out to sea. If the distance along the shore between the lifeguard and the drowning victim is 40 feet, where should the lifeguard ent er the wate r s o that he min imizes the time that it takes to reach the drowning swimm er? Solutio n: S water 50 feet A E B 0 feet 40 - L land s s w T his problem requires us to combine several of the ideas used earlier. To et up the problem let s eamine the picture and label everything. Let A be th e point on the shore closet to the lifeguard L, and let B be the point on the s ho re closest to the swimmer S. Let E be the point on the shore where the life B is 40 then 40 guard will enter the water. We are given that the distance between A and feet. Let be the distance from A to E. The distance from E to B is. We need to find a formula in terms of for the amount of time ake for the lifeguard to reach the swimmer. We use the standard it will t fo rmula D RT, in the form T, to find epressions for the time he D R pends running and for the time that he spends swimming. The total time ill be the sum of these two epressions.,

5 6 A. Time spent running: His speed (rate) is 5.7. He runs an unknown distance d from point L to point E. The Pythagorean Theorem tells us that d 0, so that d 0. Then the time he d 0 spends running from L to E is. r 5.7 B. Time spent swimming: His speed is 5.9 and the distance from E to Then the time that he spends swimming S is C. Combine these to find the total time, epressed in terms of T. The question is: for whic value is the time T at a minimum? D. Graph g 0 Y on the graphin ulator and use the minimum finder to obtain = and y = So he should enter the water approimately 4.7 feet down shore to reach the swimmer in a minimum time of calc approimately 0.89 seconds. is h In all of these problems the general strategy is to first find a formula for the quantity that you want to maim ize or mini miz e. Th is sho uld be written in terms of a single unkno wn quantity. Thi s step is the most difficult aspect of these proble ms. If the formula (equation) is quadratic then you can find the verte algebraically. However, you can always approimate the ma/min by using a graphing calculator. It is im portant to know what the two different variables of the ordered pairs on the graph represent, and what you are asked to find. It is easy to get confused in these problem s and to misinterpret the values. Section. 6 Eercises. Find the verte of the following quadratic equations and state whether it is a maimum or a minimum.. y 5 7. y 5. y y Find the m aimum or minimum value for y given the restrictions on. (R ound your answers to decimal places.) 5. y 6 6. y a. b. Solve the follo from Eample 0 < < Minimum < < 0 Maimum a. 5 < < Maimum b. 0 < < 0 Minimum wing problems. For problems 7 and 8, use the height equation. is thrown upward from the top of a 7. A ball 96-foot-high tower with an initial velocity of 80 feet per second. When does the ball reach its maimum height and how high is it at that time? 8. A ball is thrown upward from a height of 6 feet with an initial velocity of feet per second. Find its maimum height. 9. During the Civil War, the standard heavy gun for coastal artillery was the 5-inch Rodman cannon, which fired a 0-pound shell. If one of these guns is fired from the top of a 50-foot-high shoreline embankment, then the height of the shell above the water (in feet) can be approimated by the equation h , where is the horizontal distance (in feet) from the foot of the embankment to a point directly under the shell. How high does the shell go, and how far away does it hit the water? (Hint: How high will it be when it hits the water?) 0. Based on data from past years, a consultant informs Tim s Bicycles that their profit from selling bicycles is given by the equation p How much profit do they make by selling 00 4 bicycles? How many bicycles should be sold to maimize the profit?. An auto parts manufacturer makes radiators that sell for $50 each. The profit generated by selling radiators is approimated by the equation P ,000. What number of radiators will produce the largest possible profit? What is the largest possible profit?..

6 64. A farmer has 500 feet of fencing and wants to fence in a rectangular area net to a s traight river. What are the dimensions of the field with largest possible area, assuming that no fencing is needed along the river?. An open-top rectangular bo is to be made from a 5-inch by 5-inch piece of sheet metal by cut 4. ting out equal size squares from the corners and folding up the sides. What size squares should be removed in order to produce a bo with maimum volume? An open-top bo with a square base is to be constructed from 50 sq cm of material. What dimensions for the bo will produce the largest possible volume? (Hint: You need to find a formula for the height of the bo given that the surface area is 50 sq cm.) 5. Find the minimum distance from the graph of y to the point (, 0). 6. Find the minimum distance fro m the graph of y 6 to the point (, ). 7. For every point (, y) which is in the first quadrant and is on the graph y 4 4, consider the rectangle with corners (, 0), (, y), (, 0), and (, y). For which value of does this rectangle have maimal area? helicopter mile Anne miles 8. Anne is standing on a straight road and wants to reach her helicopter, whic is located miles down road from her, then a mile to the right in a field. She can run 5 miles per hour on the road and miles per hour in the field. She plans to run down the road, then cut diagonally across the field to re ach the helico pter. Where should she leav e the road in order to reach the helicopter in eactly 4 minutes? What is the least amount of time that it will take her to reach the helicopter? (Hint: See eample 4.) h