If the area of a circle is increased by a factor of 9, the circumference of that circle will be increased by what factor?
a. 3
b. 6
c. 9
d. 18
e. 81
please answer and explain
A = pi r^2
A' = 9 pi (r)^2 = pi R^2
so R^2/r^2 = 9
R/r = 3
circumference proportional to radius
so
THREE times
To answer this question, we need to understand the relationship between the area and circumference of a circle.
The area of a circle is given by the formula A = πr², where A is the area and r is the radius of the circle.
The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius of the circle.
Now, let's consider a circle where the area is increased by a factor of 9. This implies that the new area (A') is equal to 9 times the original area (A).
A' = 9A
We want to find the factor by which the circumference is increased. Let's denote this factor as 'x'. Therefore, the new circumference (C') is equal to 'x' times the original circumference (C).
C' = xC
We know that the original area (A) is equal to πr² and the original circumference (C) is equal to 2πr.
Substituting these values into the equations for A' and C', we get:
9A = πr² (equation 1)
xC = 2πr (equation 2)
To proceed, let's solve equation 1 for r²:
r² = 9A/π
Now, substitute this value of r² into equation 2:
xC = 2πr
xC = 2π√(9A/π)
xC = 2π√(9A)/√π
xC = 2√9π√A/√π
xC = 2√9A
Therefore, we can see that the new circumference (C') is equal to 2 times the square root of 9 times the original circumference (C).
This implies that x = 2√9 = 2 * 3 = 6
Therefore, the circumference of the circle will be increased by a factor of 6.
Answer: b. 6