Question

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

Answer 1

\[\sf SA ~=~31\pi~cm^2\] 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 \(\sf r^2\)

Answer 2

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

Answer 3

Do you understand how this wrks? @Kimes

Answer 4

yeah I'm just trying to rewrite everything

Answer 5

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

Answer 6

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

Answer 7

Okay.

Answer 8

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. \[\sf V = \frac{31}{3}\pi \left(\frac{\sqrt{31}}{2}\right)=
\frac{31\sqrt{31}}{6}\pi =\frac{\sqrt{31^3}}{6}\pi =\frac{31^{3/2}}{6}\pi\]

Answer 9

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

Answer 10

so i got 16. 23 cm^3

Answer 11

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

Answer 12

I'm getting \(\approx\) 90.37 cm\(^3\)

Answer 13

oh ok i just got that too

Answer 14

inputting it into your calculator, I did :

( (31\(\sf\sqrt{31}\) ) / 6 ) * \(\pi\)

Answer 15

Oh, okay. Cool!

Answer 16

thank you so much!