337 3.1 Quadratic Functions and Models 66. Volume of a Box A piece of sheet metal is 2.5 times as long as it is wide. It is to be made into a box with an open top by cutting 3-in. squares from each corner and folding up the sides. Let x represent the width (in inches) of the original piece of sheet metal. (a) Represent the length of the original piece of sheet metal in terms of x. (b) What are the restrictions on x? (c) Determine a function V that represents the volume of the box in terms of x. (d) For what values of x (that is, original widths) will the volume of the box be between 600 and 800 in.3? 67. Shooting a Free Throw If a person shoots a free throw from a position 8 ft above the floor, then the path of the ball may be modeled by the parabola y = -16x2 0.434v2 + 1.15x + 8, where v is the initial velocity of the ball in feet per second, as illustrated in the figure. (Data from Rist, C., “The Physics of Foul Shots,” Discover.) (a) If the basketball hoop is 10 ft high and located 15 ft away, what initial velocity v should the basketball have? (b) What is the maximum height of the basketball? 68. Shooting a Free Throw See Exercise 67. If a person shoots a free throw from an underhand position 3 ft above the floor, then the path of the ball may be modeled by y = -16x2 0.117v2 + 2.75x + 3. Repeat parts (a) and (b) from Exercise 67. Then compare the paths for the overhand shot and the underhand shot. 69. Sum and Product of Two Numbers Find two numbers whose sum is 20 and whose product is the maximum possible value. (Hint: Let x be one number. Then 20 - x is the other number. Form a quadratic function by multiplying them, and then find the maximum value of the function.) 70. Sum and Product of Two Numbers Find two numbers whose sum is 32 and whose product is the maximum possible value. 71. Height of an Object If an object is projected upward from ground level with an initial velocity of 32 ft per sec, then its height in feet after t seconds is given by s1t2 = -16t2 + 32t. Find the number of seconds it will take the object to reach its maximum height. What is this maximum height? 72. Height of an Object If an object is projected upward from an initial height of 100 ft with an initial velocity of 64 ft per sec, then its height in feet after t seconds is given by s1t2 = -16t2 + 64t + 100. Find the number of seconds it will take the object to reach its maximum height. What is this maximum height? 3 3 3 3 3 3 3 3 x 10 ft 15 ft 50° (x, y) 10 ft 15 ft 70° (x, y)
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