anonymous
  • anonymous
Given that the surface area of a sphere is 31 \pi cm^2, find its volume.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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Jhannybean
  • 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 \(\sf r^2\)
Jhannybean
  • Jhannybean
\[\sf 31\pi = 4\pi r^2\]\[\sf \color{red}{r^2 = \frac{31\pi}{4\pi} = \frac{31}{4}}\] \[\sf V = \frac{4}{3}\pi r^3=\frac{4}{3}\pi\color{red}{ r^2} \cdot r \]\[\sf V =\frac{4}{3} \pi \left(\frac{31}{4}\right)\cdot r = \frac{31}{3}\pi r\]
Jhannybean
  • Jhannybean
Do you understand how this wrks? @Kimes

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anonymous
  • anonymous
yeah I'm just trying to rewrite everything
Jhannybean
  • Jhannybean
Im not sure if \(\sf r\) is art of your answer, or we have to reduce it even more.
anonymous
  • anonymous
hmm its saying its wrong, so i guess keep reducing it
Jhannybean
  • Jhannybean
Okay.
Jhannybean
  • 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. \[\sf V = \frac{31}{3}\pi \left(\frac{\sqrt{31}}{2}\right)= \frac{31\sqrt{31}}{6}\pi =\frac{\sqrt{31^3}}{6}\pi =\frac{31^{3/2}}{6}\pi\]
Jhannybean
  • Jhannybean
Im 95% sure that that is your simplest, most reduced form.
anonymous
  • anonymous
so i got 16. 23 cm^3
Jhannybean
  • Jhannybean
You could leave it as \(\sf \dfrac{31\sqrt{31}}{6}\pi\) or even \(\dfrac{31^{3/2}}{6}\pi\)
Jhannybean
  • Jhannybean
I'm getting \(\approx\) 90.37 cm\(^3\)
anonymous
  • anonymous
oh ok i just got that too
Jhannybean
  • Jhannybean
inputting it into your calculator, I did : ( (31\(\sf\sqrt{31}\) ) / 6 ) * \(\pi\)
Jhannybean
  • Jhannybean
Oh, okay. Cool!
anonymous
  • anonymous
thank you so much!

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