Question

a new theater is being builtfor the city ballet. The balcony has 100 seats. the floor has 20 rows with x seats in each row. the number of people in the theater must be under 400 to meet fire regulations.

Answer 1

To determine the maximum number of seats in the theater while adhering to the fire regulations, we can set up the following equation based on the information provided:

1. **Balcony seats**: 100 seats
2. **Floor seats**: 20 rows with \( x \) seats in each row

The total number of seats in the theater can be calculated as:

\[
\text{Total seats} = \text{Balcony seats} + \text{Floor seats} = 100 + (20 \times x)
\]

According to the fire regulations, the total number of seats must be under 400:

\[
100 + (20 \times x) < 400
\]

Now, let's solve for \( x \):

1. Subtract 100 from both sides:

\[
20x < 300
\]

2. Divide both sides by 20:

\[
x < 15
\]

So, the maximum number of seats in each row on the floor can be up to 14 seats (since \( x \) must be a whole number less than 15) to keep the total seating under 400.

Thus, the configuration that meets the fire regulations is:

- 100 seats in the balcony
- Up to 14 seats per row on the floor, totaling \( 20 \times 14 = 280 \) seats on the floor

The total seating would then be:

\[
100 + 280 = 380
\]

This configuration ensures that the total number of seats is under the 400-seat limit.