Question

The lengths of two sides of a rectangle are in the ratio $2:3.$ The area of the rectangle is $864.$ What is the perimeter of the rectangle?

Answer 1

Let's call the lengths of the sides x and y, with x being the shorter side. We are given that the ratio of the sides is 2:3, or $x:y = 2:3$. The area of the rectangle is $xy = 864$.

We can set up a system of equations to represent this information:

$\begin{cases} x:y=2:3 \ xy=864 \end{cases}$

To solve this system, we can first eliminate the ratio between the sides by setting them equal:

$2x = 3y$

Then, we can substitute this expression for $y$ into the second equation to solve for $x$:

$xy = 864$

$2x(3y) = 864$

$6x^2 = 864$

$x^2 = 144$

$x = 12$

Since $x$ is the shorter side, the longer side is $y = 3x = 3(12) = 36$. The perimeter of the rectangle is $2x + 2y = 2(12) + 2(36) = 96$.