3. The side lengths of a rectangle are in the ratio \( 3: 2 \). a. If the perimeter is 15 cm , find its area. b. Another rectangle has the same perimeter, but its side lengths are in the ratio 1:5. What is the area of this rectangle?

To solve the problem, we will break it down into two parts: part (a) and part (b). ### Part (a) Let the side lengths of the rectangle be \( 3x \) and \( 2x \), where \( x \) is a common multiplier. The perimeter \( P \) of a rectangle is given by the formula: \[ P = 2(\text{length} + \text{width}) = 2(3x + 2x) = 2(5x) = 10x \] We know the perimeter is 15 cm, so we can set up the equation: \[ 10x = 15 \] Now, solve for \( x \): \[ x = \frac{15}{10} = 1.5 \] Now we can find the side lengths: - Length = \( 3x = 3(1.5) = 4.5 \) cm - Width = \( 2x = 2(1.5) = 3 \) cm Next, we calculate the area \( A \) of the rectangle: \[ A = \text{length} \times \text{width} = 4.5 \times 3 = 13.5 \text{ cm}^2 \] ### Part (b) For the second rectangle, let the side lengths be \( 1y \) and \( 5y \), where \( y \) is another common multiplier. The perimeter of this rectangle is also given by: \[ P = 2(\text{length} + \text{width}) = 2(1y + 5y) = 2(6y) = 12y \] Since the perimeter is the same (15 cm), we set up the equation: \[ 12y = 15 \] Now, solve for \( y \): \[ y = \frac{15}{12} = \frac{5}{4} = 1.25 \] Now we can find the side lengths of the second rectangle: - Length = \( 1y = 1(1.25) = 1.25 \) cm - Width = \( 5y = 5(1.25) = 6.25 \) cm Next, we calculate the area \( A \) of this rectangle: \[ A = \text{length} \times \text{width} = 1.25 \times 6.25 = 7.8125 \text{ cm}^2 \] ### Summary of Results - The area of the first rectangle is \( 13.5 \text{ cm}^2 \). - The area of the second rectangle is \( 7.8125 \text{ cm}^2 \).