Lesson 37 Section 3.1 and Appendix H Solving Maximizing Problems 1. Read the problem, and decide what is to be maximized. 2. Write this quantity as a function of one variable. 3. Rewrite the equation in the form f(x) = ax2 + bx + c. 4. Calculate x = b2a to nd when the maximum value occurs. Substitute this value into the function to obtain the maximum value. 5. Answer the question. Use the same process for minimizing. 1. A rocket is red from a point above the ground. Its height in meters above the ground after t seconds is given by h = 10t2 + 40t + 120. Find the maximum height it reaches and the number of seconds it takes to reach that height. When does the rocket hit the ground? 2. A local store that sells deli sandwiches has a xed weekly cost of $650, and variable costs for making a sandwich are $0.90. The weekly revenue that the store collects is given by R(x) = 0:003x2 + 6x, where x is the number of deli sandwiches made and sold. a) Write a pro t function to represent the weekly pro t of the store as a function of the number of sand- wiches sold. b) How many sandwiches have to be made and sold in order to have a maximum pro t? c) What is that maximum pro t? 3. A farmer has 1200 feet of fencing to enclose a rectangular area adjacent to a river. If the side along the river is not fenced, nd the length and width of the plot that will maximize the area. What is the largest area that can be enclosed? 4. A bus on a route between two cities charges a fare of $80 per person plus $5 per person for each unsold bus seat. If the bus holds 40 passengers and if x represents the number of unsold seats, nd a) a function to represent the total revenue received for the bus trip, b) the number of unsold seats that will produce the maximum revenue, and c) the maximum revenue. 2 5. A hotel with 200 rooms is lled every night when the room rate is $90. Experience has shown that for every $5 increase in cost, 10 fewer rooms will be occupied. Let n be the number of $5 increases in room cost. a) Write an equation to represent the nightly income from rooms. b) Find the room rate that will make the income a maximum? c) What is that maximum income? 6. Write the standard equation of a parabola that has vertex (5; 4) and that passes through the point (2;23). 3
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