could you please help with number 16 Im trying to study for a math placement exam next semester 5. Suppose the objective function P = xy is subject to the constraint10x + y = 100, where x and y are real numbers.19. Minimum-surface-area box Of all boxes with a square baseand 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 isexpressed as a function of one variable x.b. Find the absolute maximum value of P subject to the given20. Maximum-volume box Suppose an airline policy states that allbaggage must be box-shaped with a sum of length, width, andheight not exceeding 108 in. What are the dimensions and volumeof a square-based box with the greatest volume under these conditions?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 isa. Eliminate the variable y from the objective function so that S isexpressed as a function of one variable x.b. Find the absolute minimum value of S subject to the givenconstraint.designed to have a volume of 16 ft. The material used to makethe base costs twice as much (per square foot) as the material inthe 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 thesions 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 isclosest to the origin?8.What two nonnegative real numbers a and b whose sum is 23maximize a + b? Minimize a? + b?What two positive real numbers whose product is 50 have thesmallest possible sum?23. Closest point on a curve What point on the parabola y = 1 – x²is closest to the point (1, 1)?9.10. Find numbers x and y satisfying the equation 3x + y = 12 suchthat the product of x and y is as large as possible.24. Minimum distance Find the point P on the curve y = x that isclosest to the point (18, 0). What is the least distance between Pand (18, 0)? (Hint: Use synthetic division.)%3D25. Minimum distance Find the point P on the line y = 3x that isclosest to the point (50, 0). What is the least distance between Pand (50, 0)?Practice Exercises11. Maximum-area rectangles Of all rectangles with a perimeter of10, 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 dimensionsin terms of P.)26. Walking and swimming A man wishes to get from an initialpoint on the shore of a circular pond with radius 1 mi to a point onthe shore directly opposite (on the other end of the diameter). Heplans to swim from the initial point to another point on the shore13. 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 fixedarea 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 themaximum and minimum times for the trip?b. If he swims at 2 mi/hr and walks at 1.5 mi/hr, what are themaximum and minimum times for the trip?c. If he swims at 2 mi/hr, what is the minimum walking speedfor 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 nearestpoint on a straight shoreline; that point is 6 mi from a restaurant onthe shore (see figure). A woman plans to row the boat straight to apoint on the shore and then walk along the shore to the restaurant.16. Pen problemsa. A rectangular pen is built with one side against a barn. Twohundred meters of fencing are used for the other three sides ofthe 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 (seefigure). What are the dimensions of each pen that minimize thea. If she walks at 3 mi/hr and rows at 2 mì/hr, at which point onthe shore should she land to mìnimize the total travel time?b. If she walks at 3 mi/hr, what is the minimum speed at whichshe must row so that the quickest way to the restaurant is torow directly (with no walking)?amount of fence that must be used?Barn1001001001004 mi21,98110APRPAGES15TLO3DMacBook Pro

Question

could you please help with number 16  Im trying to study for a math placement exam next semester 5. Suppose the objective function P = xy is subject to the constraint10x + y = 100, where x and y are real numbers.19. Minimum-surface-area box Of all boxes with a square baseand 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 isexpressed as a function of one variable x.b. Find the absolute maximum value of P subject to the given20. Maximum-volume box Suppose an airline policy states that allbaggage must be box-shaped with a sum of length, width, andheight not exceeding 108 in. What are the dimensions and volumeof a square-based box with the greatest volume under these conditions?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 isa. Eliminate the variable y from the objective function so that S isexpressed as a function of one variable x.b. Find the absolute minimum value of S subject to the givenconstraint.designed to have a volume of 16 ft. The material used to makethe base costs twice as much (per square foot) as the material inthe 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 thesions 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 isclosest to the origin?8.What two nonnegative real numbers a and b whose sum is 23maximize a + b? Minimize a? + b?What two positive real numbers whose product is 50 have thesmallest possible sum?23. Closest point on a curve What point on the parabola y = 1 – x²is closest to the point (1, 1)?9.10. Find numbers x and y satisfying the equation 3x + y = 12 suchthat the product of x and y is as large as possible.24. Minimum distance Find the point P on the curve y = x that isclosest to the point (18, 0). What is the least distance between Pand (18, 0)? (Hint: Use synthetic division.)%3D25. Minimum distance Find the point P on the line y = 3x that isclosest to the point (50, 0). What is the least distance between Pand (50, 0)?Practice Exercises11. Maximum-area rectangles Of all rectangles with a perimeter of10, 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 dimensionsin terms of P.)26. Walking and swimming A man wishes to get from an initialpoint on the shore of a circular pond with radius 1 mi to a point onthe shore directly opposite (on the other end of the diameter). Heplans to swim from the initial point to another point on the shore13. 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 fixedarea 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 themaximum and minimum times for the trip?b. If he swims at 2 mi/hr and walks at 1.5 mi/hr, what are themaximum and minimum times for the trip?c. If he swims at 2 mi/hr, what is the minimum walking speedfor 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 nearestpoint on a straight shoreline; that point is 6 mi from a restaurant onthe shore (see figure). A woman plans to row the boat straight to apoint on the shore and then walk along the shore to the restaurant.16. Pen problemsa. A rectangular pen is built with one side against a barn. Twohundred meters of fencing are used for the other three sides ofthe 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 (seefigure). What are the dimensions of each pen that minimize thea. If she walks at 3 mi/hr and rows at 2 mì/hr, at which point onthe shore should she land to mìnimize the total travel time?b. If she walks at 3 mi/hr, what is the minimum speed at whichshe must row so that the quickest way to the restaurant is torow directly (with no walking)?amount of fence that must be used?Barn1001001001004 mi21,98110APRPAGES15TLO3DMacBook Pro

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