## anonymous one year ago Given that the surface area of a sphere is 31 \pi cm^2, find its volume.

1. Jhannybean

$\sf SA ~=~31\pi~cm^2$ Surface area of a sphere = $$\sf 4\pi r^2$$ Volume of a sphere = $$\sf\dfrac{4}{3}\pi r^3$$ So if we take what we know, the SA of the sphere, and plug it into the formula for SA, we can find $$\sf r^2$$

2. Jhannybean

$\sf 31\pi = 4\pi r^2$$\sf \color{red}{r^2 = \frac{31\pi}{4\pi} = \frac{31}{4}}$ $\sf V = \frac{4}{3}\pi r^3=\frac{4}{3}\pi\color{red}{ r^2} \cdot r$$\sf V =\frac{4}{3} \pi \left(\frac{31}{4}\right)\cdot r = \frac{31}{3}\pi r$

3. Jhannybean

Do you understand how this wrks? @Kimes

4. anonymous

yeah I'm just trying to rewrite everything

5. Jhannybean

Im not sure if $$\sf r$$ is art of your answer, or we have to reduce it even more.

6. anonymous

hmm its saying its wrong, so i guess keep reducing it

7. Jhannybean

Okay.

8. Jhannybean

We already have $\sf r^2=\frac{31}{4} ~~~~\text{which can imply} \implies \color{red}{r=\sqrt{\frac{31}{4}} =\frac{\sqrt{31}}{2}}$ Plug this into what we have already found for V. $\sf V = \frac{31}{3}\pi \left(\frac{\sqrt{31}}{2}\right)= \frac{31\sqrt{31}}{6}\pi =\frac{\sqrt{31^3}}{6}\pi =\frac{31^{3/2}}{6}\pi$

9. Jhannybean

Im 95% sure that that is your simplest, most reduced form.

10. anonymous

so i got 16. 23 cm^3

11. Jhannybean

You could leave it as $$\sf \dfrac{31\sqrt{31}}{6}\pi$$ or even $$\dfrac{31^{3/2}}{6}\pi$$

12. Jhannybean

I'm getting $$\approx$$ 90.37 cm$$^3$$

13. anonymous

oh ok i just got that too

14. Jhannybean

inputting it into your calculator, I did : ( (31$$\sf\sqrt{31}$$ ) / 6 ) * $$\pi$$

15. Jhannybean

Oh, okay. Cool!

16. anonymous

thank you so much!