The volume of a sphere with radius r can be written as the function V(r)=4/3πr^3 . Determine the inverse of this function and use it to find the radius of a sphere with a volume of 50 cm^3 . Round to two decimal places.

Answer 1

To find the inverse of the function V(r) = (4/3)πr^3, we need to switch the roles of the input and output variables. Let's call the inverse function R(V), where V is the volume and R is the radius.

To find the inverse function, we can start by setting V(r) = V and solving for r.

V(r) = (4/3)πr^3

V = (4/3)πr^3

3V/4π = r^3

Taking the cube root of both sides, we get:

r = (3V/4π)^(1/3)

Now we can substitute 50 cm^3 for V to find the radius R:

R = (3(50)/4π)^(1/3)

R ≈ 2.83 cm

Therefore, the radius of a sphere with a volume of 50 cm^3 is approximately 2.83 cm.

Answer 2

is that correct?

Answer 3

Apologies for the error. Here's the correct calculation:

To find the inverse of the function V(r) = (4/3)πr^3, we need to switch the roles of the input and output variables. Let's call the inverse function R(V), where V is the volume and R is the radius.

To find the inverse function, we can start by setting V(r) = V and solving for r.

V(r) = (4/3)πr^3

V = (4/3)πr^3

To isolate r, we divide both sides by (4/3)π and then take the cube root:

r^3 = V / ((4/3)π)

r = (V / ((4/3)π))^(1/3)

Now we can substitute 50 cm^3 for V to find the radius R:

R = (50 / ((4/3)π))^(1/3)

Calculating this, we get:

R ≈ 2.73 cm

Therefore, the radius of a sphere with a volume of 50 cm^3 is approximately 2.73 cm.