Question

The volume of this sphere is 4500m³.
Find the length of the radius, r.

Answer 1

`r=\frac{15\sqrt[3]{\pi ^2} }{\pi }` meters

`r=10.24` meters

Explanation:

Here is the equation for the volume of a sphere

`V=(4)/(3) \pi r^(3)`

Lets solve for
`r`.

Multiply both sides of the equation by
`(1)/((4)/(3)\pi)`.

`(1)/((4)/(3)\pi)((4)/(3)*(\pi r^3))=(1)/((4)/(3)\pi)V`

Simplify the left side.

Combine
`(4)/(3)` and
`\pi`.

`(1)/((4\pi )/(3))((4)/(3)*(\pi r^3))=(1)/((4)/(3)\pi)V`

Multiply the numerator by the reciprocal of the denominator.

`(3)/(4\pi ) ((4)/(3)*(\pi r^3))=(1)/((4)/(3)\pi)V`

Cancel the common factor of
`\pi` by factoring it out of the first term.

`(3)/(4) ((4)/(3)*r^3)=(1)/((4)/(3)\pi)V`

Combine
`(4)/(3)` and
`r^3`.

`(3)/(4)*(4r^3)/(3)=(1)/((4)/(3)\pi)V`

Multiply the numerators on the left side of the equation. Then multiply the denominators.

`(12r^3)/(4*3)=(1)/((4)/(3)\pi)V`

`(12r^3)/(12)=(1)/((4)/(3)\pi)V`

Cancel the common factor of 12.

`r^3=(1)/((4)/(3)\pi)V`

Simplify the right side.

Combine
`(4)/(3)` and
`\pi`.

`r^3=(1)/((4\pi )/(3))V`

Multiply the numerator by the reciprocal of the denominator.

`r^3=(3V)/(4\pi )`

Take the cube root of both sides of the equation to eliminate the exponent on the left side.

`r=\sqrt[3]{(3V)/(4\pi ) }`

Now we have an equation to find the radius.

Substituting our number for volume gives us

`r=\sqrt[3]{(3*4500)/(4\pi ) }`

Enter this into a calculator

`\sqrt[3]{(3*4500)/(4\pi ) }`

You can leave the answer in simplest radical form or as a decimal.

`r=\frac{15\sqrt[3]{\pi ^2} }{\pi }`

`r=10.24`

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