Question

if the area of a square has increased by a factor of 144, by how much has each side increased? each side of the square has increased by a factor of \boxed{}.

Explanation:

Step1: Recall the area formula of a square

The area \( A \) of a square with side length \( s \) is given by \( A = s^2 \).

Step2: Let the original side length be \( s_1 \) and the new side length be \( s_2 \)

Let the original area be \( A_1 = s_1^2 \) and the new area be \( A_2 = s_2^2 \). We know that the area has increased by a factor of 144, so \( \frac{A_2}{A_1}=144 \).
Substituting the area formulas, we get \( \frac{s_2^2}{s_1^2}=144 \).

Step3: Solve for the ratio of side lengths

Taking the square root of both sides of the equation \( \frac{s_2^2}{s_1^2}=144 \), we have \( \frac{s_2}{s_1}=\sqrt{144} \).
Since \( \sqrt{144} = 12 \), the side length has increased by a factor of 12.

Answer:

12