Mathematical Modeling and Optimization Problems Answers

Transcription

1 MATH& 141 Mathematical Modeling and Optimization Problems Answers 1. You are designing a rectangular poster which is to have 150 square inches of tet with -inch margins at the top and bottom of the poster and -inch margins on each side of the poster (see figure). Find a function that models the area of the entire poster A (in square inches) in terms of the width W (in inches); include the domain of this application. Math is wonderful and fun. Math is spectacularly cool and neat. Math is really wonderful and fun. I wished everyone loved mathematics as much as I do. It's a magnificient fun subject to study at all levels of school. It is so wonderful and fun. Math is wonderful and can be understood by everyone. Math is wonderful and fun. Math is spectacularly cool and so fun. Math is really wonderful and fun. I wish everyone loved mathematics as much as I do. It's a magnificient fun subject to study at all levels of school. It is so wonderful and fun. Math is wonderful and can be understood by everyone. Math is wonderful and fun. Math is spectacularly cool and so fun. Math is really wonderful and fun. I wish everyone loved mathematics as much as I do. It's a magnificient fun subject to study at all levels of school. It is so wonderful and fun. Math is wonderful and can be understood by everyone. Math is wonderful and fun. Math is spectacularly cool and so fun. Math is really wonderful and fun. I wish everyone loved mathematics as much as I do. It's a magnificient fun subject to study at all levels of school. It is so wonderful and fun. Math is wonderful and can be understood by everyone. Math is wonderful and fun. Math is spectacularly cool and so fun. Math is really fabulous and I can answer anything. interesting questions using the power of mathematical analysis. You can do it!!! Answer: 150 A 6W W 4 or A 6W 16 W W 4, 4 W W. A manufacturer wants to make a right cylindrical can (with top and bottom) with a capacity of 000 cubic centimeters (i.e. liters). Find a function that models the surface are S (in square inches) in terms of the can s radius r (in inches); include the domain of this application. Answer: A 4000 r r or r 4000 A, 0 r r. At noon a sailboat is 0 kilometers directly south of a freighter. The sailboat is traveling east at 0 kilometers per hour and the freighter is traveling south at 40 kilometers per hour. Find a function that models the distance between the ships D (in miles) in terms of the amount of time since noon t (in hours); include the domain of this application. Answer: D 0 40t 0t or D t t , 0 t

2 4. A bo with a lid is to be constructed from a square sheet of cardboard 50 inches on a side by cutting along the dotted lines as shown in the figure. The cardboard is then folded up to form the ends and sides, and the flap is folded over to form the lid. Find a function that models the volume of the bo V (in cubic inches) in terms of its depth (in inches); include the domain of this application. Answer: V 50 5 or V 5 or V , An apartment building is built in the shape of a bo with a square cross-section ( by ) and a triangular prism forming the roof. The total height of the building is 0 feet, and the sum of the length and the width of this building is 100 feet. See figure. Assuming that the height of the triangular prism forming the roof roof must be at least 6 feet (so that most of building maintenance employees can stand-up and walk through the center of that part of the building without having to bend over), find a function that models the volume of the apartment building V (in cubic feet) in terms of (in feet); include the domain of this application F.Y.I. The volume of a triangular prism equals the product (area of triangular base)(length of prism). Hence, 1 the volume V of a triangular prism of width b, height h and length L is V bhl b h L Answer: 1, 0 4 V NOTE: Since 0 represents the height or the triangular roof, which must be no less than 6 (as per the height restriction requirement given in the problem), then 0 6 4, where 4 ensures that the house s length, 100 is positive.

3 6. A closed rectangular container with a square base is to have a volume of,50 cubic inches. The material for the top and bottom of the container will cost $ per square inch, and the material for the sides will cost $ per square inch. Find a function that models the cost of the bo C (in dollars) of the bo in terms of the length of a side of the base (in inches); include the domain of this application. Answer: 7000 C 4 or C, 0 7. A rectangle is inscribed in a semicircle of radius 8 inches as shown in the figure. Find a function that models the area A of the rectangle in terms of its height h. Include the domain of this application h A h 8 Answer: A h 64 h, 0h 8 HINT: Use the Pythagorean Theorem to find half of the rectangle s length. 8. An open bo with locking tabs is to be made from a square piece of material 4 inches on a side. This is to be done by cutting squares inches by inches from each of the corners, cutting four incisions of length inches, and then folding along the dashed lines shown in the figure. Find a function that models the volume of the bo V (in cubic inches) of the bo in terms of its depth (in inches); include the domain of this application. Answer: V or V 81 6 V , 0 6

4 9. Find a function that models the length of the ladder L (in feet) that will reach from the ground, over a wall 8 feet high, to the side of a building 1 foot behind the wall, in terms of the distance between the base of the ladder and the building (in feet). Observe from the figure that length L = L 1 + L ). Include the domain of this application. Answer: L 8 1 or L, 0 HINT: Use properties associated with similar triangles and the Pythagorean Theorem. 10. A storage tank for butane gas is to be built in the shape of a right circular cylinder of altitude L feet, with a half-sphere attached to each end ( represents the radius of each half-sphere and it is measured in feet). The tank's volume is to be 144 cubic feet. See figure. l L feet Find a function that models the tank s (eterior) surface area (in square feet) in terms of the tank s radius (in feet); include the domain of this application. Answer: S 4 88 or 144 4, 0 4 S 4

5 11. Delbert stands at the top of a 00-foot cliff and throws his Algebra book upward, releasing the book 4 feet above the cliff top with a speed of 0 feet per second. Recall: s f t 16t v t s gives the vertical position s (in feet) of an 0 0 object t seconds after it is launched with an initial velocity of v 0 feet per second from an initial vertical position s0 feet presuming that once launched the object s motion is affected only by gravity. (a) Eactly how high above the ground is Delbert s book at the instant it stops going up and begins its descent? (b) Eactly how long will it take Delbert's book to hit the ground at the bottom of the cliff? Answers: 1 (a) Delbert's book will be eactly 10 4 feet above the base of the cliff at the instant it stops going up and begins its descent ( 5 seconds after Delbert releases it). That is, the book will be 1 10 above the cliff-top seconds after Delbert throws it. 8 (b) Delbert's book hits the ground at the bottom of the cliff after eactly (NOTE: This is approimately 5 seconds after he throws it). seconds. 1. A dragon kite calls for two graphite sticks as shown in the figure with the total length of the epensive sticks budgeted to 100 inches. What dimensions yield a kite with maimum surface area? 1 Answer: The surface area of the kite will be maimized for inches and y 0 inches. That is, the kite's surface area will be as large as possible if it is in the shape of a semi-circle with radius 1 inches. y A y y. This function of two independent variables can be rewritten as a function of one independent variable (either or y) by using the fact that the total length of the sticks is 100 inches. Specifically, solve y 100 for either variable and then substitute the resulting epression into the area function HINT: With A representing the total area of the kite (in square inches), 1. A y Choosing to solve the equation y 100 for y, we get y100. Substituting into

6 1 1 1 yields 1 A f , 100 where lengths and y must be nonnegative, and so 0. Use quadratic function tools to find the maimum value of A f 100 on the interval 0. A y 1. A builder is attaching a deck to the back of a house and plans to put two trees in the corners (see figure). For design reasons, the builder decides to make the square corner cutouts around the trees one third the width of the deck. If the builder has 40 feet of railing, what dimensions should the deck have in order to maimize its area and what is the deck s maimum area? House y Answer: The dimensions that will yield the maimum deck area of 180 square feet are feet and y 16 feet eactly. FIGURE NOT TO SCALE 14. An athletic facility consists of a rectangular region with a semicircle on each end (see figure). The inside perimeter of the running track is to be 00 meters while the enclosed rectangular region is to be a playing field. Determine the maimum possible area of the rectangular region; specify. Also, specify the dimensions of the playing field ( and y ) and the radius of the semicircular ends for the athletic facility whose enclosed rectangular playing field contains the largest possible area. y Answer: The athletic facility should be constructed as described below in order to obtain a 5000 rectangular playing field with the largest area possible square meters : Length is 50 meters, 100 Width is y meters (which is approimately meters), Radius of each semicircular end is 50 meters (which is approimately 16 meters)

7 15. One thousand feet of chain-link fence is to be used to construct si rectangular animal cages (as shown in the figure below). Determine the EXACT dimensions ( and y) that maimize the total enclosed area. 500 Answer: The dimensions of the region containing all si rectangular cages should be 166 feet 1 and y 15 feet in order to enclose the largest area possible ( 0,8 square feet). HINT: With A representing the area of the entire enclosed region (measured in square feet), A y. This function of two independent variables can be rewritten as a function of one independent variable (either or y) by using the fact that the total length of the sticks is 100 inches. Specifically, solve 4y 1000 for either variable and then substitute the resulting epression into the area function A y. Choosing to solve the equation 4y 1000 for y, we get 1000 y. Substituting into A y yields A f 50, where lengths and y must be nonnegative, and so 0. Use quadratic function tools to find the maimum value of f on the interval 0.

8 16. A wire 10 meters long is to be cut into two pieces. One piece will be shaped as an equilateral triangle, and the other piece will be shaped as a circle. Let represent the length of a side of the triangle. Determine the value of that maimizes the total area enclosed by both the triangle and the circle Answer: Each side of the triangle should be (which is the same as meters in length. Note that this says that should be approimately.08 meters in order to maimize the total enclosed area. ) A SOLUTION: wire 10 meters After the wire has been cut into two pieces, an equilateral triangle is made with one of the pieces and a circle is made with the other piece. With representing the length (in meters) of of a side of the triangle, then the piece of wire that is used to create the equilateral triangle must be meters long. Consequently, 10 represents the length (in meters) of the of the piece of wire that is used to create the circle. 10 Observe that only those values of in the interval 0 make sense in this application ( 0 10 corresponds to the case where all 10 meters of wire go into making the circle; corresponds to the case where all 10 meters of wire go into making the triangle). 10 wire 10 meters

9 Let A represent the combined area enclosed within the equilateral triangle and the circle (in squre meters). Then, A Area of triangle + Area of circle 1 baseheight radius. h circumeference is 10 h 10 wire 10 meters With h representing the triangle s height (in meters), then by the Pythagorean Theorem we see that 1 h h it cannot be negative; thus, h 4 h 4 h. Since h represents a height, Consequently, the area of the triangle is The area of the circle is radius. Since the the circumference of a circle is radius we have previously deduced that the circle s circumference is 10, then radius and since radius. Consequently, the area of the circle, 10 radius, can be written as. Hence, the total area enclosed is A Area of triangle + Area of circle 1 baseheight radius Observe that A is a quadratic function; consequently, we can find the EXACT 4 values associated with its optimum value. We are seeking to maimize A and we should only be looking at the parabola on the interval 0 10.

10 10 4 A 10 A 4 4 A A A A a function of the form f a b c The first coordinate of the verte of the parabola associated with A b is a

11 17. A window is to be built in the shape of a rectangle with an equilateral triangle on top. The frame that is to surround the entire window, by running along all its outer edges, is to be built with 6 meters of gold-leaf molding. What dimensions will allow the most light to pass through the window? y Answer: The rectangular portion of the window should be 1 11 meters wide and 15 meters high; the triangular portion 11 of the window should have a base that is 1 meters long 11 and its height should be 6 meters. 11 HINT: With A representing the total area of the entire window (in square meters), A y. 4 This function of two independent variables can be rewritten as a function (either or y) by using the fact that the total perimeter of the window is 6 meters. Specifically, solve y 6 for either variable and then substitute the resulting epression into the area function A y 4. 6 Choosing to solve the equation y 6 for y, we get y. Substituting into A y yields A, where lengths and y must be nonnegative, and so 0. Use quadratic function tools to find the maimum value of A f 4 on the interval 0.

12 18. The U.S. Post Office has the following restriction on sending packages through its regular mail service: the package's girth (i.e. distance around its thickest part) plus the length of the longest side of the package cannot eceed 108 inches. Suppose you want to ship a rectangular bo that is of the shape shown in the figure here. It is a bo with a square base (of side inches) and its longest side (i.e. its height) measures y inches. Determine the dimensions that yield a bo with the largest surface area possible. That is, determine the values of and y so that the package's girth plus the length of its longest side is 108 inches and its surface area is maimized. y girth Answer: The dimensions of a rectangular bo that satisfy the Post Office s restrictions and that yield a bo of maimum surface area, in 7 are inches and y 46 inches. 7 7 HINT: With A representing the total surface area of the bo (in square inches), A 4y. This function of two independent variables can be rewritten as a function (either or y) by using the fact that the total perimeter of the window is 6 meters. Specifically, solve 4 y 108 for either variable and then substitute the resulting epression into the area function A y 4. Choosing to solve the equation 4 y 108 for y, we get y Substituting into A A , where lengths and y must be 4 y yields nonnegative, and so 0 7. Use quadratic function tools to find the maimum value of A 14 4 on the interval An open-topped rectangular conduit for carrying water (i.e. a gutter) is made by folding up the edges of a sheet of aluminum 1 inches wide through an angle of 90. See figure. What (eact) depth will provide maimum cross-sectional area and hence allow the most water to flow? 1 in. The cross-sectional area we seek to maimize Answer: A conduit that is inches deep (and 6 inches wide) will allow the most water to flow (since its cross-sectional area will be as large as possible; it will be 18 square inches).

13 HINT: With A representing the representing the cross-sectional area of the conduit (in square inches), A 1 1. Use quadratic function tools to find the maimum value of A 1 on the interval 0 1. The cross-sectional area to be maimized 1 0. A zookeeper needs to add a rectangular outdoor pen to an animal house with a corner notch, as shown in the figure below. If 85 meters of fence is available for this project, what eact dimensions of the pen will maimize its area? What is the pen s maimum area? Note: No fence will be used along the walls of the animal house. Animal House 5 m 10 m Outdoor Pen Answer: The outdoor pen will have the largest area possible (namely 65 square meters) when it is a square of side 5 meters.

14 A SOLUTION: Let A represent the area of the pen (in square meters). Let represent the width of the pen (in meters). Let y represent the length of the pen (in meters). See figure below. Animal House 10 m 10 5 m y 5 Outdoor Pen y We want to find the maimum realistic value of A. A y, where y y (since 85 meters of fencing is to be used around the perimeter). y y 100 y 50. Although we can solve y 50 for either or y, I choose to solve this equation for y. Thus, y50. Substituting y50 into A y A 50. Since the area function is a QUADRATIC function, we can find the EXACT optimum value, but first we must determine the domain of this application (so we ll know where to look for this etreme value)., we get the function of one variable: Since represents a length, it cannot be negative; therefore, 0. Since 10 represents a length, it cannot be negative: therefore, Since y represents a length, it cannot be negative; therefore, y 0. But y50, and so y Since y 5 represents a length, it cannot be negative; therefore, y5 0 y 5. But y50, and so y

15 Since all four lengths described at the bottom of the previous page cannot be negative, then all four of the conditions about must hold. That is, 0 AND 10 AND 50 AND 45. The only values of which meet all four of these conditions are those in the interval Therefore, we need to look for the maimum value of the quadratic function A 50 interval (we should NOT be looking anywhere else) A f, A f, A f, on the The verte of the parabola associated with the graph b 50 of A f has 5 as its a 1 first coordinate. Observe that 5 IS in the domain of this application since 5 is in the interval Thus, the maimum value of A does occur at the verte of the parabola; specifically, A is maimized when 5. When 5 meters, y meters, and A y square meters. CONCLUSION The outdoor pen will have the largest area possible (namely 65 square meters) when it is a square of side 5 meters. Animal House 10 m 15 m 5 m 0 m Outdoor Pen 5 m 5 m

16 1. Suppose that you have 0 yards of fencing with which to build three adjacent rectangular sections as shown in the figure. Find the dimensions so that the total enclosed area is as large as possible. Answer: The enclosed area is as large as possible when the dimensions are 7½ yards by 55 yards. 7½ yards 55 yards HINT: Letting W and L represent the width and length (in yards) of the entire rectangular region, then the objective here is to maimize the total area (in square yards) A LW, where L4W 0. Choosing to solve L4W 0 for L, and then substituting this epression of L110 W into A LW, then the objective of this problem becomes to find the location of the verte on the graph 55 of A 110 W W for 0 W 55 ; this occurs at W or 7½.. You are going to enclose the region shown in the figure below with 00 feet of fencing. You will not put fencing anywhere along the barn. Determine the dimensions ( and y) that maimize the total enclosed area. Barn 5 y Answer: The dimensions of the region containing all si rectangular cages should be 5 feet and y 5 feet in order to enclose the largest area possible (704 square feet).

17 A SOLUTION Given: 00 feet of fence is to be used to enclose the entire region shown in the figure below. 5 Barn 5 y y Want: The EXACT dimensions ( and y, measured in feet) that maimize the total enclosed area. Let A represent the area of the entire enclosed region (measured in square feet). Then, A y (where the area is a function of TWO independent variables ( and y)). Since we were told that 00 feet of fencing is to be used to enclose the entire region (which includes the barn), but that none of the fencing will be used along any of the barn s sides, we see that y 5 y 00 y One must decide whether to solve y 8 00 for or for y (either choice will work). Choosing to solve y 8 00 for y: y y 8 00 y 08 y Substituting y104 into A y, we see that A f 104 Since, 5, y, and y represent lengths, then 5 and y Thus, we want to maimize the function involving ONE independent variable, A f 104 on the interval , A f is a quadratic function, we can determine the first Since coordinate of its verte as follows: b a 1.

18 Since the first coordinate of the verte assoicated with the graph of A f 104 is 5, a value in the interval (the practical domain of this application), then we conclude that the area function is as large as possible when 5. Since the other dimension of the enclosure was given by y, where y104, then with 5, we see that y F.Y.I. Although you weren t asked to determine the actual value of the largest area possible, it could easily be done now since A y, where 5 and y 5 : A square feet. NOTE: If you choose to rewrite A y as a function of y, then you d solve y 8 00 for 104 y so that substitution would yield A g y You dould then look for the maimum value of A g y 104 y y y 104y. over the interval y 99. You d find that the maimum value of A occurs at the verte of has coordinates ya, 5, 704. A f y y 104y which CONCLUSION: The dimensions of the region containing all si rectangular cages should be 5 feet and y 5 feet in order to enclose the largest area possible (704 square feet).. A farmer has 80 yards of fencing available to enclose a region in the shape of two adjoining squares with sides of length and y, respectively. See figure. Determine the nonzero values of and y so that the area of the entire enclosed region is as small as possible when all 80 yards of fencing is used. y Answer: The area of the two adjoining square regions is as small as possible (0 square yards) when 80 yards of fencing is used to enclose both the larger region that is 16 yards by 16 yards and the smaller region that is 8 yards by 8 yards.

19 HINT: With A representing the entire field s (in square yards), A y y A y This function of two independent variables can be rewritten as a function (either or y) by using the fact that the total fencing used to enclose the field is 80 yards. Specifically, solve y y y y 4 y 80 for either variable and then substitute the 80 resulting epression into the area function A y. Choosing to solve the equation 4y 80 for y, we get y40. Substituting into A y yields A 40 A , where lengths and y must be nonnegative, and so 0 0. Use quadratic function tools to find the maimum value of A on the interval You have 100 inches of wire. You want to use the wire to create two identical circles and a square. The result might look like this: (a) What should the radius of the circles be so that the total combined area of the circles and the square is as small as possible? Round your answer to the nearest ten thousandth (i.e. report four digits after the decimal point). Answer: The radius is approimately 4.86 units (rounded to the nearest ten thousandth). (b) What should the radius of the circles be so that the total combined area of the circles and the square is as large as possible? Answer: The radius should be zero.

20 5. A rectangular area is to be fenced against an eisting wall. The three sides of the fence must be 1,050 feet long. Find the dimensions of the maimum area that can be enclosed and state this maimum area. 1 Answer: The dimensions are 6 feet by 55 feet for a maimum area of 17,81 square feet. Some Math Humor