Question

the side length of the square shown is tripled. 3 cm which percent of increase is greater: the percent of increase for the perimeter of the square or the percent of increase for the area? how much greater? the percent of increase for the is greater.

Explanation:

Step1: Calculate original perimeter and area

The original side - length of the square $a = 3$ cm. The original perimeter $P_1=4a = 4\times3=12$ cm, and the original area $A_1=a^{2}=3^{2}=9$ cm².

Step2: Calculate new side - length

The side - length is tripled, so the new side - length $a_2 = 3\times3 = 9$ cm.

Step3: Calculate new perimeter and area

The new perimeter $P_2=4a_2=4\times9 = 36$ cm, and the new area $A_2=a_2^{2}=9^{2}=81$ cm².

Step4: Calculate percent increase in perimeter

The percent increase formula is $\text{Percent Increase}=\frac{\text{New Value}-\text{Original Value}}{\text{Original Value}}\times100\%$. For the perimeter, $\text{Percent Increase}_P=\frac{P_2 - P_1}{P_1}\times100\%=\frac{36 - 12}{12}\times100\%=\frac{24}{12}\times100\% = 200\%$.

Step5: Calculate percent increase in area

For the area, $\text{Percent Increase}_A=\frac{A_2 - A_1}{A_1}\times100\%=\frac{81 - 9}{9}\times100\%=\frac{72}{9}\times100\% = 800\%$.

Step6: Find the difference in percent increase

The difference in percent increase is $800\%-200\% = 600\%$.

Answer:

The percent of increase for the area is greater. It is 600% greater.