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.