a sphere has a volume of 500/3pi cubvic centimeters. What is the total surface area, in square centimeters, of the square? The answer is 100pi, but i don't know why. HELP!!!!!!!!!

You just have to relate the formulas for volume and area.

v = 4/3 pi r^3

use that to get r.

500pi/3 = 4pi/3 r^3

500pi/3 * 3/4pi = r^3
125 = r^3
r = 5

Knowing that r=5, we can now get the area:

a = 4pi r^2 = 4pi * 25 = 100pi

To find the surface area of a sphere, we can use the formula:

Surface Area = 4πr²

where r is the radius of the sphere.

First, let's find the radius of the sphere using the given volume. The volume of a sphere is given by the formula:

Volume = (4/3)πr³

Given that the volume is 500/3π cubic centimeters, we can set up the equation:

500/3π = (4/3)πr³

To find r, we need to solve for it. We can start by multiplying both sides of the equation by 3/4π to cancel out the fractions:

r³ = (500/3π) * (3/4π)

Simplifying this expression, we get:

r³ = 500/4

Next, we can take the cube root of both sides of the equation to isolate r:

r = (500/4)^(1/3)

Now that we have the radius, we can substitute it back into the surface area formula to find the total surface area:

Surface Area = 4πr²

Surface Area = 4π * ((500/4)^(1/3))²

Simplifying further:

Surface Area = 4π * (500/4)^(2/3)

Surface Area = 4π * (125)^(2/3)

Using the exponent rule, we can rewrite (125)^(2/3) as the cube root of (125^2):

Surface Area = 4π * ∛(125^2)

Since 125 can be written as 5^3, we substitute it:

Surface Area = 4π * ∛(5^6)

Now, since the cube root and the square are inverse operations, we can rewrite it as:

Surface Area = 4π * (√(5^6))^3/2

Simplifying:

Surface Area = 4π * (5^3)

Surface Area = 4π * 125

Surface Area = 500π

So the total surface area of the sphere is 500π square centimeters.

The answer 100π you mentioned may be a typo or an error. The correct surface area is 500π.