Question

The telephone company offers two billing plans for local calls. Plan 1 charges ​$30 per month for unlimited calls and Plan 2 charges ​$11 per month plus ​$0.04 per call.
a. Use an inequality to find the number of monthly calls for which Plan 1 is more economical than Plan 2.
b. Explain the meaning of the answer to part a.

Answer 1

To solve the problem, we need to set up the cost equations for both plans and find the point at which Plan 1 becomes more economical than Plan 2.

Let \( x \) be the number of calls made in a month.

**Plan 1**: The cost is a constant $30 per month, regardless of the number of calls.
- Cost for Plan 1: \( C_1 = 30 \)

**Plan 2**: The cost consists of a base fee of $11 plus $0.04 for each call made.
- Cost for Plan 2: \( C_2 = 11 + 0.04x \)

Now, we want to find when Plan 1 is more economical than Plan 2, so we set up the inequality:

\[
30 < 11 + 0.04x
\]

Now, solving for \( x \):

1. Subtract 11 from both sides:
\[
30 - 11 < 0.04x
\]
\[
19 < 0.04x
\]

2. Divide both sides by 0.04:
\[
\frac{19}{0.04} < x
\]
\[
475 < x
\]

This translates to:
\[
x > 475
\]

### a. The inequality is:
\[
x > 475
\]

### b. Explanation of the answer:
The answer \( x > 475 \) means that if you make more than 475 calls in a month, then Plan 1 (which has a flat rate of $30) will be more economical compared to Plan 2 (which charges a base fee plus $0.04 per call). Conversely, if you make 475 or fewer calls, Plan 2 would be the more economical choice. This information helps consumers decide which plan to choose based on their expected volume of calls for that month.