01 May For what x does the function 9(z) – 10 + 40: — x2 have its maximum value? 2. Find
For what x does the function 9(z) – 10 + 40: — x2 have its maximum value? 2. Find the maximum value of the function /(x) = 12x —x2, and give the value of x where this maximum occurs. 3. Find the minimum value of 1(t) = t3 —612 + 40, t > 0. and give the value of t where this minimum occurs. 4. For what t does the function 1(t) = i2 – 24t have its minimum value? 5. Optimization with Constraint Find the maximum of Q = xy if x+y= 2. 8. Oplimizedion with Constraint Find two positive numbers x and y that maximize Q = x2y if z + y = 2. 7. Optimization with Ccogroint Find the minimum of Q = x2 +y2 ifx+y= 6. 8. In Exercise 7, can there be a maximum for Q = x2 + y2 if x + y = 6? Justify your answer. 9. Minimizing a Sum Find the positive values of x and y that minimize S x + y if xy = 36, and find this minimum value. 10. Maximizing a Product Find the positive values of x, y, and z that maximize Q = xyz, if z + y = 1 and y + z = 2. What is this maximum value? 11. Area There are $320 available to knee in a rectangular garden. The fencing for the side of the garden facing the road costs S6 per foot and the fencing for the other throe sides costs S2 per foot. Nee Fig. 12(a).) Consider the problem of finding the dimensions of the largest possible garden. (a) Determine the objective and constraint equations. (b) Express the quantity to be maximized as a function of z. (c) Find the optimal values of and p. 12. Volume Figure 12(b) shows an open rectangular box with a square base. Consider the problem of finding the val-ues of x and h for which the volume is 32 cubic feet and the total surface area of the box is minimal. (The sur-face area is the sum of the areas of the five faces of the box.) (a) Determine the objective and constraint equations. (b) Express the quantity to be minimized as a function of z.
(c) Find the optimal values of x and h.
Figure 12
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13. Volume Postal requirements specify that parcels must have length plus girth of at most 84 inches. Consider the problem of finding the dimensions of the square-ended rectangular package of greatest volume that is mailable. (a) Draw a square-ended rectangular box. Label each edge of the square end with the letter x and label the remaining dimension of the box with the letter tt. (b) Express the length plus the girth in terms of x and h. (o) Determine the objective and constraint equations. (d) Express the quantity to be maximized as a function of x. (a) Find the optimal values of x and h. 14. Perimeter Consider the problem of finding the dimensions of the rectangular garden of area 100 square meters for which the amount of fencing needed to surround the gar-den is as small as possible. (a) Draw a picture of a rectangle and select appropriate letters for the dimensions. (b) Determine the objective and constraint equations. (c) Find the optimal values for the dimensions. 15. Cost A rectangular garden of area 75 square feet is to be surrounded on three sides by a brick wall costing S10 per foot and on one side by a fence costing 85 per foot. Mod the dimensions of the garden that minimize the cost of materials. 16. Cost A closed rectangular box with a square base and a volume of 12 cubic feet is to be constructed from two dif-ferent types of materials. The top is made of a metal cost-ing 82 per square foot and the remainder of wood costing SI per square foot. Find the dimensions of the box for which the cost of materials is minimized.
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