Question

the area of a rectangle is ( 375 \text{in}^2 ). the ratio of the length to the width is ( 5:3 ). find the length and the width... the length of the rectangle is ( 15 ) in.

Explanation:

Step1: Define variables with ratio

Let the length be \( 5x \) and the width be \( 3x \) (since the ratio of length to width is \( 5:3 \)).

Step2: Use area formula for rectangle

The area of a rectangle is \( \text{length} \times \text{width} \). So, \( (5x)(3x)=375 \).
Simplify the left side: \( 15x^{2}=375 \).

Step3: Solve for \( x \)

Divide both sides by 15: \( x^{2}=\frac{375}{15}=25 \).
Take the square root of both sides: \( x = 5 \) (we take the positive root as length and width are positive).

Step4: Find length and width

Length: \( 5x = 5\times5 = 25 \) in.
Width: \( 3x = 3\times5 = 15 \) in. (Wait, the original wrong answer had length as 15, but correct length is 25, width is 15. Let's re - check.
Wait, area is \( 5x\times3x = 15x^{2}=375\), so \( x^{2}=25\), \( x = 5 \). Then length is \( 5\times5 = 25 \), width is \( 3\times5 = 15 \). So the correct length is 25 in, width is 15 in.

Answer:

The length of the rectangle is \( \boldsymbol{25} \) in, and the width of the rectangle is \( \boldsymbol{15} \) in.