Problem 9 - Algebra and Trigonometry

Question

You are choosing between two health clubs. Club A offers membership for a fee of \(\$ 40\) plus a monthly fee of \(\$ 25 .\) Club \(\mathrm{B}\) offers membership for a fee of \(\$ 15\) plus a monthly fee of \(\$ 30\). After how many months will the total cost at each health club be the same? What will be the total cost for each club?

Step-by-Step Solution

Answer

The total cost at each health club will be the same after 5 months, and the total cost for each club at this time will be $165.

Step 1: Formulate the Cost Equations for both Clubs

The total cost at Club A per month is given by the equation \(C_A = 40 + 25m\), where \(m\) is the number of months. Similarly, we can write the total cost at Club B as \(C_B = 15 + 30m\).

Step 2: Solve for m, where \(C_A = C_B\)

We set the cost of the memberships equal to each other to find the number of months where they will cost the same. So, we solve for \(m\) in the equation \(40 + 25m = 15 + 30m\). Simplifying this, we get \(5m = 25\), and solving for \(m\) gives us \(m = 5\) months.

Step 3: Compute the total cost at each club

Substituting \(m = 5\) months into either the cost equation for Club A or Club B will give the total cost when both memberships cost the same. So, the total cost will be \(C_A = C_B = 40 + 25 * 5 = $165\).