Question

3) The cost of a gym membership can be represented by 25 + 5x, where x is the number of visits to the gym.

a. How many times can you visit if you have a $100 budget?


b. What is the y-intercept? What does it mean in context?


c. What is the slope? What does it mean in context?

Answer 1

Let's break down the cost function for a gym membership, represented by the equation \( C(x) = 25 + 5x \), where \( C(x) \) is the total cost and \( x \) is the number of visits to the gym.

### a. How many times can you visit if you have a $100 budget?

To find the number of visits allowed within a $100 budget, we set the equation equal to 100 and solve for \( x \):

\[
100 = 25 + 5x
\]

Subtract 25 from both sides:

\[
100 - 25 = 5x \\
75 = 5x
\]

Now, divide both sides by 5:

\[
x = \frac{75}{5} \\
x = 15
\]

Therefore, you can visit the gym **15 times** with a $100 budget.

### b. What is the y-intercept? What does it mean in context?

The y-intercept of the equation \( C(x) = 25 + 5x \) is found by setting \( x = 0 \):

\[
C(0) = 25 + 5(0) = 25
\]

So, the y-intercept is **25**.

**In context**, this means that the initial cost of the gym membership (before any visits are made) is $25. This could represent a sign-up fee or a base membership cost.

### c. What is the slope? What does it mean in context?

The slope of the equation \( C(x) = 25 + 5x \) is **5**.

**In context**, the slope represents the cost per visit to the gym. Specifically, for each visit to the gym, you incur an additional cost of **$5**.