Question

7. Suppose a sphere has a volume of 3,000 cm3. Find the surface area.

The volume of a sphere: V= 4/3 pi r^3

Part I: Find the radius of the sphere by setting the volume equal to the formula above. Show your work and round the radius to the nearest hundredth.

Part II: Use the radius you found in Part I with the surface area formula below to the find the surface area of the sphere. Show your work and round your answer to the nearest tenth.

The surface area of a sphere: SA= 4 pi r^2

Answer 1

Part 1

`\boxed{r = 8.9 \;cm}`

Part 2

`\boxed{SA=995.4\;cm^2}`

Explanation:

If r is the radius of a sphere

Volume of a sphere:

`V = (4)/(3)\pi r^2`

Surface Area

`SA = 4\pi r^2`

Part 1
We are given volume = 3,000 cm³

Setting this to volume to the equation for volume gives

`3000 = (4)/(3) \pi r^3\\\\`

Multiplying both sides by 3/4 yields

`(3)/(4)\cdot 3000 = \pi r^3\\\\2250 = \pi r^3\\\\r^3 = (2250)/(\pi)\\\\r^3 \approx 716.1972\\\\\\r = \sqrt[3]{716.1972} \\\\r = 8.947\\\\\\\textrm{Rounded to the nearest tenth:}\\\\\boxed{r = 8.9 \;cm}\\\\\text{(Answer\;Part 1)}\\`

Part 2

Using r = 8.9 cm we can compute the surface area SA using the formula for surface area

`SA = 4\pi r^2\\\\SA = 4\pi (8.9)^2\\\\SA = 995.38\;cm^2\\\\\textrm{Rounded to the nearest tenth:}\\\\\boxed{SA=995.4\;cm^2}\\\\\textrm{(Answer Part 2)}`