Review Exercise Set 16

Transcription

1 Review Execise Set 16 Execise 1: A ectangula plot of famland will be bounded on one side by a ive and on the othe thee sides by a fence. If the fame only has 600 feet of fence, what is the lagest aea that the fame can enclose? What ae the dimensions of the ectangula plot? Execise : You have been assigned the task of designing a containe in the fom of a ight cicula cylinde that can hold a volume of 1000 cubic centimetes. If the cost of the mateial to be used fo the sides costs $0.01 pe squae centimete and the cost of the mateial fo the top and bottom costs $0.0 pe squae centimete, find the adius, height, and cost of the least expensive containe?

2 Execise : A 10 inch by 15 inch piece of cadboad will be used to constuct a box with a lid. Two equal squaes will be emoved fom one side and two equal ectangles ae emoved fom the othe cones so that the emaining cadboad can be folded to fom the box. Find the value of x that will maximize the volume of the box. Execise 4: A etail stoe puchases computes fom a manufactue. The etail stoe's cuent annual demand is 1875 computes. It costs the etaile $ to stoe a compute in its inventoy and the ode costs ae $50. What is the optimum numbe of computes the etaile should puchase in each ode and how often must the odes be made?

3 Review Execise Set 16 Answe Key Execise 1: A ectangula plot of famland will be bounded on one side by a ive and on the othe thee sides by a fence. If the fame only has 600 feet of fence, what is the lagest aea that the fame can enclose? What ae the dimensions of the ectangula plot? Make dawing of the poblem Define the dimensions in a single vaiable We know that the total fencing available is 600 feet, so the sum of the thee sides of fence must equal 600. y + y + x = 600 y + x = 600 y = x y = x Setup the equation to be optimized We need to find the lagest aea that can be enclosed so we would use the equation fo the aea of a ectangle. A = xy A(x) = x(100-1 x) Detemine the limitations on function The sides of the ectangle must be nonnegative so the function is bound by the following inequalities: x x 0 1 x 100 x 600 So we must find the absolute maximum of A(x) on the closed inteval [0, 600].

4 Execise 1 (Continued): Find the fist deivative A(x) = x(100-1 x) A(x) = 100x - 1 x A'(x) = x Find the citical numbes 0 = x x = 100 Evaluate the equation at the citical numbe and inteval endpoints A(x) = 100x - 1 x A(0) = 100(0) - 1 (0) A(0) = 0 A(100) = 100(100) - 1 (100) A(100) = 1,690, ,000 A(100) = 845,000 A(600) = 100(600) - 1 (600) A(600) =,80,000 -,80,000 A(600) = 0 The maximum aea occus when x = 100. Find the value of the othe dimension y = x y = (100) y = y = 650 The dimensions that will maximize the aea ae 650 feet by 100 feet and the maximum aea will be 845,000 squae feet.

5 Execise : You have been assigned the task of designing a containe in the fom of a ight cicula cylinde that can hold a volume of 1000 cubic centimetes. If the cost of the mateial to be used fo the sides costs $0.01 pe squae centimete and the cost of the mateial fo the top and bottom costs $0.0 pe squae centimete, find the adius, height, and cost of the least expensive containe? Make dawing of the poblem Define the dimensions in a single vaiable V= π 1000 = π h 1000 = h π h Setup the equation to be optimized We ae looking to minimize the cost of the mateial used to ceate the cylinde, so we would need to find the suface aea of the cylinde and then multiply by the appopiate costs. SA = SA + SA + SA top side bottom = π + πh + π = π + πh 1000 = π + π π 000 = π +

6 Execise (Continued): ( ) = π ( 0.0) + ( 0.01) C 0.04π = Detemine the limitations on function > 0 Find the fist deivative ( ) ( ) C = 0.04π C = 0.08π 0 0 = 0.08π π = Find the citical numbes Set numeato and denominato equal to zeo 0.08π 0 0 = 0.08π 0 50 = π 50 = π 4. Find the height when = h = π 1000 π 17. ( 4.) =

7 Execise (Continued): Find the cost when = 4. C() = C(4.) = 0.04 (4.) + C(4.) C(4.) 6.97 The cylinde would need to have a adius of 4. centimetes and a height of 17. centimetes in ode to minimize the cost at $6.97. Execise : A 10 inch by 15 inch piece of cadboad will be used to constuct a box with a lid. Two equal squaes will be emoved fom one side and two equal ectangles ae emoved fom the othe cones so that the emaining cadboad can be folded to fom the box. Find the value of x that will maximize the volume of the box. Define the dimensions in a single vaiable We will look at the length fist The length of the base and lid must be the same so fo now we will let it be L and thee ae two lengths of x which is the height of the sides when they ae folded up. The total length of the cadboad is 15 inches so: 15 = x + L + x + L 15 = x + L 15 - x = L x = L

8 Execise (Continued): Now we will look at the width The width of the base and lid ae not known so we will let them be W fo now. The width of the cadboad has two pieces x inches long that will be cut out on the ends. The total width of the cadboad is 10 inches so: 10 = x + W + x 10 = x + W 10 - x = W Setup the equation to be optimized We ae looking to maximize the volume of the box so we would use the volume fomula fo ou equation. V = LWH V(x) = (7.5 - x)(10 - x)(x) V(x) = (75-5x + x )(x) V(x) = x - 5x + 75x Detemine the limitations on function The measuements cannot be negative so all thee ae set equal to o geate than zeo. Height Length Width x x 0 10 x 0 x 7.5 x 10 x 7.5 x 5 x is bound by the closed inteval [0, 5] Find the fist deivative V'(x) = 6x - 50x + 75

9 Execise (Continued): Find the citical numbes 0 = 6x - 50x + 75 ( 50) ( 50) 4( 6)( 75) 6 ( ) ± x = = 50 ± ± 700 = x= x= 1 1 x 1.96 x 6.7 x cannot be 6.7 because it is outside of the closed inteval [0, 5] so the citical numbe we would use is Find the second deivative V''(x) = 1x - 50 Test the second deivative at x = 1.96 to confim it is an absolute maximum V''(1.96) = 1(1.96) - 50 V''(1.96) = Since the second deivative is negative at x = 1.96 it is concave down and theefoe means that this is an absolute maximum. The volume of the box is maximized when x is 1.96 inches.

10 Execise 4: A etail stoe puchases computes fom a manufactue. The etail stoe's cuent annual demand is 1875 computes pe yea which sell at a unifom ate thoughout the yea. It costs the etaile $ pe yea to stoe a compute in its inventoy and the ode costs ae $50. What is the optimum numbe of computes the etaile should puchase in each ode to minimize the odeing and stoage costs? Also, how often ae the odes to be made? Define the vaiable(s) x = ode size (numbe of computes pe ode) 1875 = numbe of odes pe yea x x = aveage inventoy (aveage numbe of compute stoed) Setup the equation to be optimized We want to optimize the total odeing and stoage costs so ou function would be the sum of the odeing costs and stoage costs. C(x) = odeing costs + stoage costs C(x) = (numbe of odes)(cost/ode) + (aveage inventoy)(cost/item) C(x) = ( 1875 x )(50) + ( )() x C(x) = 9750 x Find the fist deivative + 1.5x C(x) = 9750x x C'(x) = -9750x C'(x) = x 1.5x 9750 C'(x) = x Find the citical numbes Set numeato and denominato equal to zeo 1.5x = 0 x = 0 1.5x = 9750 x = 0 x = 6500 x = ± 50 x cannot be zeo o negative, so the only citical numbe is 50.

11 Execise 4 (Continued): Find the second deivative C'(x) = -9750x C''(x) = x C''(x) = x Test the second deivative at x = 50 to confim it is an absolute minimum C''(50) = ( 50 ) C''(50) = 0.01 Since C''(50) is geate than zeo, this would be the location of the absolute minimum fo the cost function. Find how many odes need to be made 1875 = numbe of odes = numbe of odes