Problem 28 - College Algebra Essentials

Question

You need to rent a rug cleaner. Company A will rent the machine you need for \(\$ 22\) plus \(\$ 6\) per hour. Company \(B\) will rent the same machine for \(\$ 28\) plus \(\$ 4\) per hour. After how many hours of use will the total amount spent at each company be the same? What will be the total amount spent at each company?

Step-by-Step Solution

Answer

The total amount spent at each company will be the same after 3 hours, with the total cost being \$40.

Step 1: Define the Equations

First, formulate two linear equations to represent the cost of rental from each company. Let \( h\) be the hours of use. For Company A, the cost is \( C_A = 22 + 6h\). For Company B, the cost is \( C_B = 28 + 4h\). Our task is to find the time \( h\) at which \( C_A = C_B\).

Step 2: Equate the two functions

Set the two equations equal to each other to find the hour at which the two companies' charges intersect, i.e., \(22 + 6h = 28 + 4h\).

Step 3: Solve for h

Subtract \(4h\) from both sides of the equation and subtract \(22\) from both sides of the equation to solve for \(h\), which yields \(2h = 6\). Solving for \(h\) gives \(h = 3\).

Step 4: Calculate the total amount spent

Substitute \(h = 3\) into either of the cost functions to find the total amount spent. If we substitute into the first function, we get \(C_A = 22 + 6 * 3 = \$40\).