Question

The table below shows the cost for a movie theater membership with an initial start-up fee charged in the first month.

Number of Months
Cost ($)
1
52
52
2
74
74
3
96
96




The cost per month is $
.

The initial cost is $
.

Answer 1

To analyze the cost structure in the table provided, let's break down the information given:

1. For **1 month**, the total cost is **$52**.
2. For **2 months**, the total cost is **$74**.
3. For **3 months**, the total cost is **$96**.

From this data, we can determine the initial cost and the monthly cost:

**Step 1: Find the Initial Cost**

The cost appears to have a fixed initial start-up fee followed by a regular monthly fee. We can analyze the differences between the costs for consecutive months to find out the monthly fee.

- The cost from **1 month** to **2 months** increases from **$52** to **$74**, which is an increase of **$22**.
- The cost from **2 months** to **3 months** increases from **$74** to **$96**, which is also an increase of **$22**.

This suggests that the monthly cost is **$22**.

**Step 2: Calculate the Initial Cost**

Now that we know the monthly cost, we can calculate the initial cost:

Let \( x \) be the initial cost and \( m \) be the monthly cost.
From the data above,
- For **1 month**: \( x + m = 52 \)

Thus, with \( m = 22 \):
\[
x + 22 = 52 \implies x = 52 - 22 = 30
\]

So, the initial cost is **$30**.

**Summary:**
- The cost per month is **$22**.
- The initial cost is **$30**.