We start with a circle x. If we increase the radius by 50% then the area will increase by 100 in.². Find the original circumference of circle x. ( hint let r = oringinal radius)

I set it up as x+100= (3.14)(.50✖️R^2)

But is that right. I don't really know how to solve that

π(1.5r)^2 = πr^2 + 100

9/4 πr^2 = πr^2 + 100
5/4 πr^2 = 100
r^2 = 80/π
r = 4√(5/π)

so, the original circumference is

c = 2πr = 8√(5π)

To find the original circumference of the circle, we will first find the original radius (r), and then use the formula for the circumference of a circle: C = 2πr.

Let's start with the information given:

1. Increasing the radius by 50% means the new radius (r') is 1.5 times the original radius (r).
So, we have r' = 1.5r.

2. The increase in the area of the circle is given as 100 in².
We know that the area of a circle is A = πr², so the new area (A') is (πr')².
The difference in area is A' - A = (πr')² - πr² = 100.

Now, let's solve for the original radius (r):

1. Expand (πr')²:
(πr')² = (π(1.5r))² = (2.25πr²)

2. Substitute this into the equation A' - A = 100:
(2.25πr²) - (πr²) = 100
Simplifying further: 2.25πr² - πr² = 100
Combine like terms: (2.25 - 1)πr² = 100
Simplify: (1.25πr²) = 100

3. Solve for r²:
Divide both sides by 1.25π: r² = 100 / (1.25π)

4. Take the square root of both sides to find r:
r = √(100 / (1.25π))

Now that we have the value of r, we can calculate the original circumference (C) using the formula C = 2πr.

C = 2πr
Substituting the value of r, we get:
C = 2π * √(100 / (1.25π))

Simplifying further, we find the original circumference of circle x.