a Use the percent proportion to calculate the change in the side length.
B
b Do the exact same thing you did in Part A.
C
c What patterns did you notice in the previous parts?
A
a 21 %
B
b 21 %
C
c 21 %
Explanation: See solution.
Practice makes perfect
a It is given that the original side length is 4 meters. Since the area of a square is found by squaring its side length, we can find the area of the original square.
A_(original)= 4^2= 16
If the side length grows by 10 % , the square's dimensions change and the area of the square will increase. We will first calculate the change in the side's length by using the percent proportion.
If the square's side length increases by 0.4centimeters, then the new side length becomes 4+0.4=4.4 meters. Now we can calculate the new area.
A_(new)= 4.4^2= 19.36
The area of the square increased. To find the amount of increase, we will subtract the original area from the new area.
A_(new) -A_(original) &= 19.36- 16
& = 3.36
We now have enough information to calculate the percent increase in the area.
Percent increase=Amount of increase/Original amount
The new length of the square's side is 6+0.6=6.6 meters. Now we can calculate the new area.
A_(new)=6.6^2= 43.56
The area of the square increased. To find the amount of increase, we will subtract the original area from the new area.
A_(new) -A_(original) &= 43.56- 36
& = 7.56
We now have enough information to calculate the percent increase in the area.
Percent increase=Amount of increase/Original amount
c Instead of doing the same thing again, let's prove that a 10 % increase in the square's side length corresponds to a 21 % increase in its area no matter the side length. 10 % increase in the square's side length
⇓ ?
21 % increase in its area
For this purpose, we will call the square's side length s which means the square has an area of s^2. Let's use the percent proportion to find the change in the square's side length.
An expression for the new square's side length is 1.1s. By squaring this expression we get the new area.
A=( 1.1s)^2= 1.21s^2
The new area is 1.21s^2. From this expression, we can tell that the new area will be 21 % larger because 1.21 as a percent is 121 %.
1.21 = 121/100= 121 %
This corresponds to an increase of 21 %. Regardless of the length’s numeric value, in this case 8 centimeters, anytime the square’s side length is increased by 10 %, then the area of the square will always increase by 21 %. Let's check our answer by repeating what we did previously in Part A and B.
