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23. Area Figure 14(a) shows a Norman window, which consists of a rectangle capped by a semicircular region. Find the value of x such that the perimeter of the window will be 14 feet and the area of the window will be as large as possible. x 2x (a) Figure 14 h h (b) 2mx Side unrolled emsidor

23. Area Figure 14(a) shows a Norman window, which consists of a rectangle capped by a semicircular region. Find the value of x such that the perimeter of the window will be 14 feet and the area of the window will be as large as possible. x 2x (a) Figure 14 h h (b) 2mx Side unrolled emsidor

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Exc 2.5 Q23 needed to be solved correctly
15. Cost A rectangular garden of area 75 square feet is to be sur- rounded on three sides by a brick wall costing S10 per foot and on one side by a fence costing $5 per foot. Find the dimensions of the garden that minimize the cost of materials. 16. Cost A closed rectangular box with a square base and a vol- ume of 12 cubic feet is to be constructed from two different types of materials. The top is made of a metal costing $2 per square foot and the remainder of wood costing S1 per square foot. Find the dimensions of the box for which the cost of materials is minimized. 17. Surface Area Find the dimensions of the closed rectangular box with square base and volume 8000 cubic centimeters that can be constructed with the least amount of material. 18. Volume A canvas wind shelter for the beach has a back, two square sides, and a top. Find the dimensions for which the volume will be 250 cubic feet and that requires the least pos- sible amount of canvas. 19. Area A farmer has $1500 available to build an E-shaped fence along a straight river so as to create two identical rectangu- lar pastures. (See Fig. 13.) The materials for the side parallel to the river cost $6 per foot, and the materials for the three sections perpendicular to the river cost $5 per foot. Find the dimensions for which the total area is as large as possible. Figure 13 Rectangular pastures along a river. 20. Area Find the dimensions of the rectangular garden of great- est area that can be fenced off (all four sides) with 300 meters of fencing. 21. Maximizing a Product Find two positive numbers, x and y, whose sum is 100 and whose product is as large as possible. 22. Minimizing a Sum Find two positive numbers, x and y, whose product is 100 and whose sum is as small as possible. River 23. Area Figure 14(a) shows a Norman window, which consists of a rectangle capped by a semicircular region. Find the value of x such that the perimeter of the window will be 14 feet and the area of the window will be as large as possible. 2π.χ ndio h h (b) Side unrolled h 2x (a) Figure 14 maldor 24. Surface Area A large soup can is to be designed so that the can will hold 167 cubic inches (about 28 ounces) of soup. [See Fig. 14(b).] Find the values of x and h for which the amount of metal needed is as small as possible. 25. In Example 3 we can solve the constraint equation (2) for x instead of w to get x=20-w. Substituting this for x in (1), we get A=xw=20-ww. 1 ww. 2.5 Optimization Problems 169 Sketch the graph of the equation A = 20w -². and show that the maximum occurs when w= 20 and x 10. 26. Cost A ship uses 5x² dollars of fuel per hour when traveling at a speed of x miles per hour. The other expenses of operating the ship amount to $2000 per hour. What speed minimizes the cost of a 500-mile trip? (Hint: Express cost in terms of speed and time. The constraint equation is distance speed x time.] 27. Cost A cable is to be installed from one corner, C, of a rectan- gular factory to a machine, M, on the floor. The cable will run along one edge of the floor from C to a point, P, at a cost of $6 per meter, and then from P to M in a straight line buried under the floor at a cost of $10 per meter (see Fig. 15). Let x denote the distance from C to P. If M is 24 meters from the nearest point, A, on the edge of the floor on which lies, and A is 20 meters from C, find the value of x that minimizes the cost of installing the cable and determine the minimum cost. 20 x P Figure 16 24 Figure 15 28. Area A rectangular page is to contain 50 square inches of print. The page has to have a 1-inch margin on top and at the bottom and a -inch margin on each side (see Fig. 16). Find the dimen- sions of the page that minimize the amount of paper used. in. in. У Print Area Popt Aree Print Ares Print Area Prist Area Print Area Print Area Print Aves Print Area Prist Aren Print Area Print Area Print Area Print Aren Print Print Area M 1 in. 1 in.
Transcribed Image Text:15. Cost A rectangular garden of area 75 square feet is to be sur- rounded on three sides by a brick wall costing S10 per foot and on one side by a fence costing $5 per foot. Find the dimensions of the garden that minimize the cost of materials. 16. Cost A closed rectangular box with a square base and a vol- ume of 12 cubic feet is to be constructed from two different types of materials. The top is made of a metal costing $2 per square foot and the remainder of wood costing S1 per square foot. Find the dimensions of the box for which the cost of materials is minimized. 17. Surface Area Find the dimensions of the closed rectangular box with square base and volume 8000 cubic centimeters that can be constructed with the least amount of material. 18. Volume A canvas wind shelter for the beach has a back, two square sides, and a top. Find the dimensions for which the volume will be 250 cubic feet and that requires the least pos- sible amount of canvas. 19. Area A farmer has $1500 available to build an E-shaped fence along a straight river so as to create two identical rectangu- lar pastures. (See Fig. 13.) The materials for the side parallel to the river cost $6 per foot, and the materials for the three sections perpendicular to the river cost $5 per foot. Find the dimensions for which the total area is as large as possible. Figure 13 Rectangular pastures along a river. 20. Area Find the dimensions of the rectangular garden of great- est area that can be fenced off (all four sides) with 300 meters of fencing. 21. Maximizing a Product Find two positive numbers, x and y, whose sum is 100 and whose product is as large as possible. 22. Minimizing a Sum Find two positive numbers, x and y, whose product is 100 and whose sum is as small as possible. River 23. Area Figure 14(a) shows a Norman window, which consists of a rectangle capped by a semicircular region. Find the value of x such that the perimeter of the window will be 14 feet and the area of the window will be as large as possible. 2π.χ ndio h h (b) Side unrolled h 2x (a) Figure 14 maldor 24. Surface Area A large soup can is to be designed so that the can will hold 167 cubic inches (about 28 ounces) of soup. [See Fig. 14(b).] Find the values of x and h for which the amount of metal needed is as small as possible. 25. In Example 3 we can solve the constraint equation (2) for x instead of w to get x=20-w. Substituting this for x in (1), we get A=xw=20-ww. 1 ww. 2.5 Optimization Problems 169 Sketch the graph of the equation A = 20w -². and show that the maximum occurs when w= 20 and x 10. 26. Cost A ship uses 5x² dollars of fuel per hour when traveling at a speed of x miles per hour. The other expenses of operating the ship amount to $2000 per hour. What speed minimizes the cost of a 500-mile trip? (Hint: Express cost in terms of speed and time. The constraint equation is distance speed x time.] 27. Cost A cable is to be installed from one corner, C, of a rectan- gular factory to a machine, M, on the floor. The cable will run along one edge of the floor from C to a point, P, at a cost of $6 per meter, and then from P to M in a straight line buried under the floor at a cost of $10 per meter (see Fig. 15). Let x denote the distance from C to P. If M is 24 meters from the nearest point, A, on the edge of the floor on which lies, and A is 20 meters from C, find the value of x that minimizes the cost of installing the cable and determine the minimum cost. 20 x P Figure 16 24 Figure 15 28. Area A rectangular page is to contain 50 square inches of print. The page has to have a 1-inch margin on top and at the bottom and a -inch margin on each side (see Fig. 16). Find the dimen- sions of the page that minimize the amount of paper used. in. in. У Print Area Popt Aree Print Ares Print Area Prist Area Print Area Print Area Print Aves Print Area Prist Aren Print Area Print Area Print Area Print Aren Print Print Area M 1 in. 1 in.
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