Problem 10 - Algebra and Trigonometry

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 costs of renting from both companies will equalize after 3 hours. The total cost of renting from either company will be $40 if the machine is used for 3 hours.

Step 1: Define Variables and Formulate Equations

Let \(x\) represent the number of hours and \(y\) the total cost in dollars. The cost of renting from company A can be formulated as: \(y = 22 + 6x\). Similarly, the cost of renting from company B is expressed as: \(y = 28 + 4x\).

Step 2: Equate the Two Equations

To find when the total cost will be the same at each company, set both equations equal to each other: \(22 + 6x = 28 + 4x\).

Step 3: Solve for x

First simplify the equation by getting all terms involving \(x\) on one side and constant terms on the other: \(6x - 4x = 28 - 22\). This simplifies further to \(2x = 6\). Finally, solve for \(x\) by dividing both sides by 2: \(x = 3\). This means that the costs of renting from both companies will equalize after 3 hours.

Step 4: Find the Total Cost

Substitute \(x = 3\) into either of the two original equations to find \(y\), the total cost. Using the first equation: \(y = 22 + 6 * 3 = 40\). Thus, the total cost of renting from either company will be $40 if the machine is used for 3 hours.