Question

Two square room have a total floor area of 89m^2.one room is 3m longer each way than the other. Find the dimensions of the two rooms

Answer 1

Let the side length of the smaller square room be \( x \) meters. Consequently, the side length of the larger square room, which is 3 meters longer on each side, will be \( x + 3 \) meters.

The area of the smaller room can be expressed as:
\[
\text{Area of the smaller room} = x^2
\]

The area of the larger room can be expressed as:
\[
\text{Area of the larger room} = (x + 3)^2
\]

According to the problem, the total area of the two rooms is 89 m²:
\[
x^2 + (x + 3)^2 = 89
\]

Expanding the equation:
\[
x^2 + (x^2 + 6x + 9) = 89
\]

Combining like terms gives:
\[
2x^2 + 6x + 9 = 89
\]

Subtracting 89 from both sides results in:
\[
2x^2 + 6x + 9 - 89 = 0
\]
\[
2x^2 + 6x - 80 = 0
\]

Next, we can simplify the equation by dividing everything by 2:
\[
x^2 + 3x - 40 = 0
\]

Now we will factor the quadratic equation:
\[
(x + 8)(x - 5) = 0
\]

This gives us two possible solutions for \( x \):
\[
x + 8 = 0 \quad \Rightarrow \quad x = -8 \quad (\text{not a valid solution since dimensions can't be negative})
\]
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]

Thus, the side length of the smaller room is:
\[
x = 5 \, \text{meters}
\]

The side length of the larger room is:
\[
x + 3 = 5 + 3 = 8 \, \text{meters}
\]

In conclusion, the dimensions of the two rooms are:
- Smaller room: \( 5 \) meters x \( 5 \) meters
- Larger room: \( 8 \) meters x \( 8 \) meters