My question is about lengths and I am so lost.

Question

The length of a rectangle is 3 centimeters more than its width. If the width is increased by 2 centimeters and the length is increased by 4 centimeters, a new rectangle is formed that has an area of 44 square centimeters more than the area of the original rectangle. Find the dimensions of the original rectangle.

Answer 1

We have two rectangles in this problem. The original rectangle, which we will say has width = W₁, length = L₁, and area = A₁, and the new rectangle, which we will say has width = W₂, length = L₂, and area = A₂. Let's start with the original rectangle.

Let W₁ = x

Next, we know the length is 3 cm more than its width:

L₁ = W₁ + 3 = x + 3

The area of the rectangle can be found as follows:

A₁ = W₁*L₁ = (x)*(x+3) = x² + 3x

Next, let's look at the new rectangle. The width is being increased by 2 cm from the width of the original rectangle. We can write an equation as follows:

W₂ = W₁ + 2

W₂ = x + 2

The length is being increased by 4 cm from the length of the original rectangle. We can write an equation as follows:

L₂ = L₁ + 4

L₂ = (x + 3) + 4

L₂ = x + 7

The area of the new rectangle can be found as follows:

A₂ = W₂*L₂ = (x+2)*(x+7) = x² + 2x + 7x + 14 = x² + 9x + 14

Finally, we know the area of the new rectangle is 44 cm² larger than the area of the original rectangle. We can write an equation as follows:

A₁ + 44 = A₂

Simply plug in our expressions from above for A₁ and A₂ and solve for x:

(x² + 3x) + 44 = x² + 9x + 14, (cancel out x² on each side)

3x + 44 = 9x + 14

30 = 6x

x = 5

Now to finally solve the problem, look at our expressions for W₁ and L₁ from above and plug in our answer for x to get original dimensions of 5 cm x 8 cm.