Let's denote the width of the rectangle as \( w \) and the length as \( l \).
Given:
1. The perimeter of the rectangle is 46 cm, so we have:
\( 2w + 2l = 46 \)
2. The length is 2 cm less than \(\frac{2}{3}\) times the width, which can be expressed as:
\( l = \frac{2}{3}w - 2 \)
We can solve the system of equations to find the dimensions of the rectangle and then calculate the area.
First, let's express \( l \) in terms of \( w \) using the second given condition:
\( l = \frac{2}{3}w - 2 \)
Now we can substitute \( l \) into the formula for the perimeter:
\( 2w + 2\left(\frac{2}{3}w - 2\right) = 46 \)
\( 2w + \frac{4}{3}w - 4 = 46 \)
Multiply through by 3 to clear the fraction:
\( 6w + 4w - 12 = 138 \)
\( 10w - 12 = 138 \)
\( 10w = 150 \)
\( w = 15 \)
Now that we have the width, we can find the length:
\( l = \frac{2}{3}w - 2 = \frac{2}{3}(15) - 2 = 10 - 2 = 8 \)
So, the dimensions of the rectangle are:
Width (w) = 15 cm
Length (l) = 8 cm
The area of the rectangle is given by:
\( \text{Area} = \text{length} \times \text{width} \)
\( \text{Area} = 15 \times 8 \)
\( \text{Area} = 120 \, \text{cm}^2 \)
Therefore, the dimensions of the rectangle are 15 cm (width) and 8 cm (length), and the area of the rectangle is 120 square centimeters.