SECTION 3.4 Building Quadratic Models from Verbal Descriptions and from Data 177 What should the dimensions of the rectangle be so that the area of the rectangle is a maximum? 16. Architecture A special window has the shape of a rectangle surmounted by an equilateral triangle. See the figure. If the perimeter of the window is 16 feet, what dimensions will admit the most light? [Hint: Area of an equilateral x triangle 3 4 ,2 = where x is the length of a side of the triangle.] x x x 17. Gas Mileage Every automobile manufacturer is required to provide a fuel efficiency rating for each model car they produce. A rating of “28 mpg highway” is the estimated gas mileage while the car is driven at high speeds under normal driving conditions. The Federal Highway Administration recently performed a study to compare the speed x (in miles per hour) with the gas mileage y (in miles per gallon) of a particular automobile. The results are shown in the table. Speed mpg 15 22.3 20 25.5 25 27.5 30 29.0 35 28.8 40 30.0 45 29.9 50 30.2 55 30.4 60 28.8 65 27.4 70 25.3 75 23.3 (a) Use a graphing utility to draw a scatter plot of the data. Comment on the type of relation that may exist between the two variables. (b) Use a graphing utility to find the quadratic function of best fit that models the relation between speed and gas mileage. (c) Use a graphing utility to graph the function found in part (b), and approximate the speed at which the gas mileage is greatest. (d) With a graphing utility, graph the quadratic function of best fit on the scatter plot. 10. Enclosing the Most Area with a Fence A farmer with 2000 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed? 11. Suspension Bridge A suspension bridge with weight uniformly distributed along its length has twin towers that extend 75 meters above the road surface and are 400 meters apart. The cables are parabolic in shape and are suspended from the tops of the towers.The cables touch the road surface at the center of the bridge. Find the height of the cables at a point 100 meters from the center. (Assume that the road is level.) 12. Architecture A parabolic arch has a span of 120 feet and a maximum height of 25 feet. Choose suitable rectangular coordinate axes and find the equation of the parabola. Then calculate the height of the arch at points 10 feet, 20 feet, and 40 feet from the center. 13. Constructing Rain Gutters A rain gutter is to be made of aluminum sheets that are 12 inches wide by turning up the edges 90 .° See the illustration. (a) What depth will provide maximum cross-sectional area and allow the most water to flow? (b) What depths will allow at least 16 square inches of water to flow? x x x x 12 in. 12 2 2x 14. Norman Windows A Norman window has the shape of a rectangle surmounted by a semicircle of diameter equal to the width of the rectangle. See the figure. If the perimeter of the window is 20 feet, what dimensions will admit the most light (maximize the area)? Credit: Geno EJ Sajko Photography/ Shutterstock 15. Constructing a Stadium A track-and-field playing area is in the shape of a rectangle with semicircles at each end. See the figure.The inside perimeter of the track is to be 1500 meters.
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