To solve the given problem, let's break it down step-by-step.
1. Understand the problem: We are given the perimeter of the rectangle, which is 40 yards, and we know that the length of the rectangle is three times its width. We need to find the area of the rectangle.
2. Define the variables:
- Let the width of the rectangle be [tex]\( w \)[/tex].
- According to the problem, the length of the rectangle is [tex]\( 3w \)[/tex].
3. Write the formula for the perimeter of a rectangle:
[tex]\[
\text{Perimeter} = 2 \times (\text{length} + \text{width})
\][/tex]
Substitute the given values:
[tex]\[
40 = 2 \times (3w + w)
\][/tex]
4. Simplify the equation:
[tex]\[
40 = 2 \times 4w
\][/tex]
[tex]\[
40 = 8w
\][/tex]
5. Solve for [tex]\( w \)[/tex]:
[tex]\[
w = \frac{40}{8}
\][/tex]
[tex]\[
w = 5
\][/tex]
6. Find the dimensions of the rectangle:
- Width [tex]\( w = 5 \)[/tex] yards
- Length [tex]\( 3w = 3 \times 5 = 15 \)[/tex] yards
7. Calculate the area of the rectangle:
[tex]\[
\text{Area} = \text{length} \times \text{width}
\][/tex]
[tex]\[
\text{Area} = 15 \times 5
\][/tex]
[tex]\[
\text{Area} = 75 \text{ square yards}
\][/tex]
The area of the rectangle is [tex]\( 75 \)[/tex] square yards.
