5. Suppose the objective function P = xy is subject to the constraint 10x + y = 100, where x and y are real numbers. 19. Minimum-surface-area box Of all boxes with a square base and a volume of 8 m, which one has the minimum surface area? (Give its dimensions.) a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x. b. Find the absolute maximum value of P subject to the given 20. Maximum-volume box Suppose an airline policy states that all baggage must be box-shaped with a sum of length, width, and height not exceeding 108 in. What are the dimensions and volume of a square-based box with the greatest volume under these condi tions? constraint. Suppose S x + 2y is an objective function subject to the con- straint xy = 50, for x > 0 and y > 0. 6. 21. Shipping crates A square-based, box-shaped shipping crate is a. Eliminate the variable y from the objective function so that S is expressed as a function of one variable x. b. Find the absolute minimum value of S subject to the given constraint. designed to have a volume of 16 ft. The material used to make the base costs twice as much (per square foot) as the material in the sides, and the material used to make the top costs half as much (per square foot) as the material in the sides. What are the dimen- 7. What two nonnegative real numbers with a sum of 23 have the sions of the crate that minimize the cost of materials? largest possible product? 22. Closest point on a line What point on the line y = 3x + 4 is closest to the origin? 8. What two nonnegative real numbers a and b whose sum is 23 maximize a + b? Minimize a? + b? What two positive real numbers whose product is 50 have the smallest possible sum? 23. Closest point on a curve What point on the parabola y = 1 – x² is closest to the point (1, 1)? 9. 10. Find numbers x and y satisfying the equation 3x + y = 12 such that the product of x and y is as large as possible. 24. Minimum distance Find the point P on the curve y = x that is closest to the point (18, 0). What is the least distance between P and (18, 0)? (Hint: Use synthetic division.) %3D 25. Minimum distance Find the point P on the line y = 3x that is closest to the point (50, 0). What is the least distance between P and (50, 0)? Practice Exercises 11. Maximum-area rectangles Of all rectangles with a perimeter of 10, which one has the maximum area? (Give the dimensions.) 12. Maximum-area rectangles Of all rectangles with a fixed perim- eter of P, which one has the maximum area? (Give the dimensions in terms of P.) 26. Walking and swimming A man wishes to get from an initial point on the shore of a circular pond with radius 1 mi to a point on the shore directly opposite (on the other end of the diameter). He plans to swim from the initial point to another point on the shore 13. Minimum-perimeter rectangles Of all rectangles of area 100, which one has the minimum perimeter? and then walk along the shore to the terminal point. 14. Minimum-perimeter rectangles Of all rectangles with a fixed area A, which one has the minimum perimeter? (Give the dimen- sions in terms of A.) a. If he swims at 2 mi/hr and walks at 4 mi/hr, what are the maximum and minimum times for the trip? b. If he swims at 2 mi/hr and walks at 1.5 mi/hr, what are the maximum and minimum times for the trip? c. If he swims at 2 mi/hr, what is the minimum walking speed for which it is quickest to walk the entire distance? 15. Minimum sum Find positive numbers x and y satisfying the equa- tion xy = 12 such that the sum 2x + y is as small as possible. 27. Walking and rowing A boat on the ocean is 4 mi from the nearest point on a straight shoreline; that point is 6 mi from a restaurant on the shore (see figure). A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. 16. Pen problems a. A rectangular pen is built with one side against a barn. Two hundred meters of fencing are used for the other three sides of the pen. What dimensions maximize the area of the pen? b. A rancher plans to make four identical and adjacent rectan- gular pens against a barn, each with an area of 100 m² (see figure). What are the dimensions of each pen that minimize the a. If she walks at 3 mi/hr and rows at 2 mì/hr, at which point on the shore should she land to mìnimize the total travel time? b. If she walks at 3 mi/hr, what is the minimum speed at which she must row so that the quickest way to the restaurant is to row directly (with no walking)? amount of fence that must be used? Barn 100 100 100 100 4 mi 21,981 10 APR PAGES 15 TLO 3D MacBook Pro