Question

1. if you double the width of a square, how much does the area of the square increase? $a = s^2$

Explanation:

Step1: Define original square variables

Let original side length = $s$. Original area: $A_1 = s^2$

Step2: Define modified square variables

Double the width (side length): new side = $2s$. New area: $A_2 = (2s)^2 = 4s^2$

Step3: Calculate area increase factor

Find ratio of new to original area: $\frac{A_2}{A_1} = \frac{4s^2}{s^2} = 4$

Step4: Find absolute increase value

Area increase = $A_2 - A_1 = 4s^2 - s^2 = 3s^2$

Answer:

The area becomes 4 times the original area, which is an increase of $3s^2$ (or a 300% increase relative to the original area).