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:
- Balcony seats: 100 seats
- 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 \( x \):
- Subtract 100 from both sides:
\[ 20x < 300 \]
- 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.
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