Question

A rectangle with dimensions of 4 units by 5 units is enlarged by a scale factor of 1.2. By what percent does its area increase?

Answer 1

The area of any shape after dilation is equal to the area of the original shape multiplied by the square of the scale factor:

$A_{\text{new}}=k^2\cdot A_{\text{original}}$

Where

A_new indicates the area of the shape after the dilation

A_original is the area of the original shape

k is the scale factor

The first step is to determine the area of the original shape and the new shape, using the formula:

$A=w\cdot l$

The original shape has dimensions 4 and 5, so its area is:

$\begin{gathered} A_{\text{original}}=4\cdot5 \\ A_{\text{original}}=20 \end{gathered}$

Next is to determine the area of the shape after the dilation (A_new)

The scale factor is k=1.2

$\begin{gathered} A_{\text{new}}=k^2\cdot A_{\text{original}} \\ A_{\text{new}}=(1.2)^2\cdot20 \\ A_{\text{new}}=1.44\cdot20 \\ A_{\text{new}}=28.8 \end{gathered}$

Now that we calculated both areas, we can determine the increase percentage.

- First, calculate the increase (I), which is the difference between the area of the new shape and the area of the original shape:

$\begin{gathered} I=A_{\text{new}}-A_{\text{original}} \\ I=28.8-20 \\ I=8.8 \end{gathered}$

-Second, divide the increase by the original area and multiply the result by 100

$\begin{gathered} \frac{I}{A_{\text{original}}}\cdot100 \\ \frac{8.8}{20}\cdot100 \\ 0.44\cdot100=44 \end{gathered}$

This means that the area increased 44% after the dilation