The radius of a circle is increased by 85%. How much will the area increase in percent? Round your answer to the nearest tenth if a percent.
I'm stuck on this question. Thanks
-Zach
original area = pi(1^2) = pi
new area = pi(1.85^2) = 3.4225pi
increase = 3.4225pi - pi = 2.4225pi
percentage increase = 2.4225pi/pi = 2.4225
= 242.3 %
To find out how much the area of a circle increases when the radius is increased by a certain percentage, we need to understand the relationship between the radius and the area of a circle.
The formula for the area of a circle is given by: A = π * r^2, where A represents the area and r represents the radius.
Let's consider the initial radius as r and the initial area as A.
When the radius is increased by 85%, the new radius can be calculated as r + 0.85 * r.
To find the new area (let's call it A'), we substitute the new radius into the area formula: A' = π * (r + 0.85 * r)^2.
To calculate how much the area increases in percentage, we need to compare A' and A.
Let's begin by simplifying the expression A'.
A' = π * (r + 0.85 * r)^2
= π * (1.85r)^2
= π * 3.4225r^2
= 3.4225 * π * r^2
Now, we can compare A' and A to determine how much the area increased.
The increase in area is: ΔA = A' - A
ΔA = (3.4225 * π * r^2) - (π * r^2)
= (2.4225 * π * r^2)
To find the increase as a percentage, we divide ΔA by the initial area A and multiply by 100.
Percentage increase in area = (ΔA / A) * 100
= ((2.4225 * π * r^2) / (π * r^2)) * 100
= (2.4225 * 100)
= 242.25%
Therefore, the area will increase by approximately 242.25% when the radius is increased by 85%. Rounded to the nearest tenth, the area would increase by 242.3%.
I hope this explanation helps! Let me know if you have any more questions.