Question

A rectangular restaurant kitchen has an area of 220 square meters. Its perimeter is 62 meters. What are the dimensions of the kitchen?

meters by
meters

Answer 1

To find the dimensions of the rectangular kitchen, we can use the formulas for area and perimeter of a rectangle. Let’s denote the length as \( L \) and the width as \( W \).

1. **Area Equation**:
\[
L \times W = 220 \quad \text{(1)}
\]

2. **Perimeter Equation**:
\[
2L + 2W = 62 \quad \text{(2)}
\]
Simplifying equation (2):
\[
L + W = 31 \quad \text{(3)}
\]

Now, we can solve these two equations. From equation (3), we can express \( W \) in terms of \( L \):
\[
W = 31 - L \quad \text{(4)}
\]

Next, we substitute equation (4) into equation (1):
\[
L \times (31 - L) = 220
\]
Expanding this:
\[
31L - L^2 = 220
\]
Rearranging gives us a standard quadratic equation:
\[
L^2 - 31L + 220 = 0
\]

Now we can solve for \( L \) using the quadratic formula:
\[
L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = -31 \), and \( c = 220 \).

Calculating the discriminant:
\[
b^2 - 4ac = (-31)^2 - 4 \times 1 \times 220 = 961 - 880 = 81
\]

Now substituting it into the formula:
\[
L = \frac{31 \pm \sqrt{81}}{2}
\]
\[
L = \frac{31 \pm 9}{2}
\]
Calculating the two possible values for \( L \):
1. \( L = \frac{40}{2} = 20 \)
2. \( L = \frac{22}{2} = 11 \)

So the dimensions are \( L = 20 \) meters and \( W = 11 \) meters (using equation (4) to get \( W = 31 - L \)).

Thus, the kitchen dimensions are:
\[
20 \text{ meters by } 11 \text{ meters}
\]

### Final Answer
The dimensions of the kitchen are:
**20 meters by 11 meters**.