Problem 28 - College Algebra

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

After 3 hours of use, the total cost spent at each company will be the same and this cost will be $40.

Step 1: Formulate the cost equations

Let's say the number of hours the machine is rented is \(x\). The cost to rent from Company A can be formulated as \(A(x) = 22 + 6x\), while for Company B, it is \(B(x) = 28 + 4x\).

Step 2: Solve the equation

Find the value of \(x\) when \(A(x) = B(x)\). Set the two equations equal to each other i.e., \(22 + 6x = 28 + 4x\). Solve this equation for \(x\). For that, first subtract \(4x\)' from each side to get \(2x = 6\), and then divide each side by 2 to get \(x = 3\). So, the costs are equal when the machine is rented for exactly 3 hours.

Step 3: Find the total cost

To find the total cost at each company, substitute \(x = 3\) into either \(A(x)\) or \(B(x)\) (because they are equal at this point). For instance, \(A(3) = 22 + 6 * 3 = 40\). Therefore, the total cost spent at each company will be $40 when the machine is rented for 3 hours.