A rectangle has side lengths in the ratio 2:3. The ratio of the rectangles's perimeter to its area is 5:9. What is the length of the longer leg of the rectangle?

User Icon for Reiny Reiny answered
11 years ago

let the width be 2x and the length be 3x

Area = 6x^2
perimeter = 10

10x/6x^2 = 5/9
30x^2 = 90x

divide by 30 and factor
x(x-3) = 0
x = 0 or x = 3, x=0 is not admissable (we have to have some sort of a width)

the width is 6 and the length is 9

check:
area = 54
perimeter = 30
what is 30/54 ??

User Icon for Explain Bot Explain Bot answered
11 months ago

To solve this problem, we can use algebra.

Let's assume that the shorter side of the rectangle has a length of 2x, and the longer side has a length of 3x.

The perimeter of a rectangle is given by the formula: P = 2l + 2w, where l and w are the lengths of the longer and shorter sides, respectively.

So, the perimeter of this rectangle can be written as: P = 2(2x) + 2(3x) = 4x + 6x = 10x.

The area of a rectangle is given by the formula: A = l * w. In this case, the area will be: A = (2x)(3x) = 6x^2.

The problem states that the ratio of the rectangle's perimeter to its area is 5:9. Therefore, we can write the following equation:

(P / A) = 5 / 9

Substituting the values we found:

(10x / 6x^2) = 5 / 9

To solve this equation, we can cross-multiply:

9 * 10x = 5 * 6x^2

90x = 30x^2

Now, divide both sides of the equation by 30x:

90 / 30 = 30x^2 / 30x

3 = x

Since the longer side of the rectangle has a length of 3x, the length of the longer side is 3 * 3 = 9.

Therefore, the length of the longer leg of the rectangle is 9.