251 2.4 Linear Functions 79. Projectile Motion A ball thrown straight upward from a height of 5 ft with an initial velocity of 64 ft per sec has height h1t2 feet after t seconds, where h1t2 = -16t2 + 64t + 5. What is the ball’s average velocity (that is, its average rate of change in height) during the first second? 80. Velocity of a Helicopter A helicopter is rising straight upward. Its distance from the ground t seconds after takeoff is s1t2 feet, where s1t2 = t2 + t. What is the helicopter’s average velocity (that is, its average rate of change in height) during the first 4 sec? (Modeling) Cost, Revenue, and Profit Analysis A firm will break even (no profit and no loss) as long as revenue just equals cost. The value of x (the number of items produced and sold) where C1x2 = R1x2 is the break-even point. Assume that each of the following can be expressed as a linear function. Find (a) the cost function, (b) the revenue function, and (c) the profit function. (d) Find the break-even point and determine whether the product should be produced, given the restrictions on sales. See Example 10. Fixed Cost Variable Cost Price of Item 81. $ 500 $ 10 $ 35 No more than 18 units can be sold. 82. $2700 $150 $280 No more than 25 units can be sold. 83. $1650 $400 $305 All units produced can be sold. 84. $ 180 $ 11 $ 20 No more than 30 units can be sold. (Modeling) Break-Even Point The manager of a small company that produces roof tile has determined that the total cost in dollars, C1x2, of producing x units of tile is given by C1x2 = 200x + 1000, and that the revenue in dollars, R1x2, from the sale of x units of tile is given by R1x2 = 240x. 85. Find the break-even point and the cost and revenue at the break-even point. 86. Suppose the variable cost is actually $220 per unit, instead of $200. How does this affect the break-even point? Is the manager better off or not? 87. Use the first two points in the table to find the slope of the line. 88. Use the second and third points in the table to find the slope of the line. 89. Make a conjecture by filling in the blank: If we use any two points on a line to find its slope, we find that the slope is ________ in all cases. 90. Find the distance between the first two points in the table. (Hint: Use the distance formula.) Relating Concepts For individual or collaborative investigation (Exercises 87–96) The table shows several points on the graph of a linear function. Work Exercises 87–96 in order, to see connections between the slope formula, distance formula, midpoint formula, and linear functions. x y 0 -6 1 -3 2 0 3 3 4 6 5 9 6 12
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