Exc 2.5Q15 and Q19 needed to be solved correctly in 30 minutes and get the thumbs up please show me neat and clean work for it 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 footand on one side by a fence costing $5 per foot. Find thedimensions 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 differenttypes of materials. The top is made of a metal costing $2 persquare foot and the remainder of wood costing S1 per squarefoot. Find the dimensions of the box for which the cost ofmaterials is minimized.17. Surface Area Find the dimensions of the closed rectangularbox with square base and volume 8000 cubic centimeters thatcan be constructed with the least amount of material.18. Volume A canvas wind shelter for the beach has a back, twosquare sides, and a top. Find the dimensions for which thevolume 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 fencealong a straight river so as to create two identical rectangu-lar pastures. (See Fig. 13.) The materials for the side parallelto the river cost $6 per foot, and the materials for the threesections perpendicular to the river cost $5 per foot. Find thedimensions for which the total area is as large as possible.dFigure 13 Rectangularpastures 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 metersof 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, whoseproduct is 100 and whose sum is as small as possible.River23. Area Figure 14(a) shows a Norman window, which consistsof a rectangle capped by a semicircular region. Find the valueof x such that the perimeter of the window will be 14 feet andthe area of the window will be as large as possible.2πχhh(b)Side unrolled2x(a)Figure 14maldor24. Surface Area A large soup can is to be designed so that thecan will hold 167 cubic inches (about 28 ounces) of soup. [SeeFig. 14(b).] Find the values of x and h for which the amountof metal needed is as small as possible.h25. In Example 3 we can solve the constraint equation (2) for x insteadof w to get x = 20-w. Substituting this for x in (1), we getA = xw = 20 --ww.-1/w)w.2.5 Optimization Problems 169Sketch the graph of the equationA = 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 ata speed of x miles per hour. The other expenses of operatingthe ship amount to $2000 per hour. What speed minimizes thecost of a 500-mile trip? (Hint: Express cost in terms of speedand 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 runalong 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 buriedunder the floor at a cost of $10 per meter (see Fig. 15). Let xdenote the distance from C to P. If M is 24 meters from thenearest point, A, on the edge of the floor on which lies, andA is 20 meters from C, find the value of x that minimizes thecost of installing the cable and determine the minimum cost.20PFigure 1624Figure 1528. 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 bottomand 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.yPrint Area Popt Aree Print AresPrint Area Prist Area Print AreaPrint Area Print Area Print AreaPrist Aron Print Area Print AreaPrint AreaM1 in.1 in.

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15. Cost A rectangular garden of area 75 square feet is to be sur- rounded on three sides by a brick wall costing $10 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