Problem 80 - College Algebra Essentials

Question

A tennis club offers two payment options. Members can pay a monthly fee of \(\$ 30\) plus \(\$ 5\) per hour for court rental time. The second option has no monthly fee, but court time costs \(\$ 7.50\) per hour. a. Write a mathematical model representing total monthly costs for each option for \(x\) hours of court rental time. b. Use a graphing utility to graph the two models in a \([0,15,1]\) by \([0,120,20]\) viewing rectangle. c. Use your utility's trace or intersection feature to determine where the two graphs intersect. Describe what the coordinates of this intersection point represent in practical terms. d. Verify part (c) using an algebraic approach by setting the two models equal to one another and determining how many hours one has to rent the court so that the two plans result in identical monthly costs.

Step-by-Step Solution

Answer

The mathematical models for total monthly costs for each option are \(C1 = 30 + 5x\) and \(C2 = 7.5x\). From the graph, the two plans cost the same when \(x = 12\) hours, meaning that if a member rents the court for less than 12 hours, the first membership option is cheaper, otherwise, the second option is cheaper. We verified this point of intersection algebraically.

Step 1: Writing Mathematical Models

Let \(x\) represent court rental time in hours. Then we can write the total monthly cost for the first option as \(C1 = 30 + 5x\) and for the second option as \(C2 = 7.5x\).

Step 2: Graphing the Models

Use a graphing utility to graph the two models. The graph of the first function \(C1 = 30 + 5x\) is a straight line with slope of 5 and y-intercept of 30. The graph of the second function \(C2 = 7.5x\) is also a straight line, but with a slope of 7.5 and a y-intercept of 0. Use a viewing rectangle with x-values ranging from 0 to 15 and y-values ranging from 0 to 120.

Step 3: Finding the Intersection Point

Use the graphing utility's trace or intersection feature to determine where the two graphs intersect. This is the point where the two payment plans have the same cost.

Step 4: Verifying with an Algebraic Approach

Set \(30 + 5x = 7.5x\) and solve for \(x\). This gives us \(x = 30 / (7.5 - 5) = 12\) hours, which confirms that for 12 hours of court rental time, both membership options result in identical monthly costs.