Question

A rectangular lawn has an area of 187 square meters. Its perimeter is 56 meters. What are the dimensions of the lawn?

meters by
meters

Answer 1

Let's denote the length of the rectangular lawn as \( l \) meters and the width as \( w \) meters.

We have two equations based on the information provided:

1. The area of the rectangle:
\[
l \times w = 187
\]

2. The perimeter of the rectangle:
\[
2l + 2w = 56
\]
This simplifies to:
\[
l + w = 28
\]

Now we have a system of equations:

\[
\begin{align*}
1. & \quad lw = 187 \quad \text{(1)} \\
2. & \quad l + w = 28 \quad \text{(2)}
\end{align*}
\]

From equation (2), we can express \( w \) in terms of \( l \):
\[
w = 28 - l
\]

Now we substitute \( w \) into equation (1):
\[
l(28 - l) = 187
\]
Expanding this gives:
\[
28l - l^2 = 187
\]
Rearranging the equation, we get:
\[
l^2 - 28l + 187 = 0
\]

Next, we'll use the quadratic formula \( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 1 \), \( b = -28 \), and \( c = 187 \).
Calculating the discriminant:
\[
b^2 - 4ac = (-28)^2 - 4 \cdot 1 \cdot 187 = 784 - 748 = 36
\]

Now we can find \( l \):
\[
l = \frac{28 \pm \sqrt{36}}{2} = \frac{28 \pm 6}{2}
\]

Calculating the two possible values for \( l \):
\[
l = \frac{34}{2} = 17 \quad \text{and} \quad l = \frac{22}{2} = 11
\]

So, we have two possible dimensions for \( l \):
- If \( l = 17 \), then \( w = 28 - 17 = 11 \)
- If \( l = 11 \), then \( w = 28 - 11 = 17 \)

Thus, the dimensions of the lawn are:
\[
\text{11 meters} \times \text{17 meters}
\]

So the answer is:
**11 meters by 17 meters**.