the surface area of a sphere is 75 square centimeters,
what is the volume of the sphere in cubic cm?
Formula:
The volume inside a sphere is: \(V = \frac43\pi r^3 \qquad V= ?\).
The surface area of a sphere is: \(A = 4\pi r^2 \qquad A=75\ cm^2\).
\(\begin{array}{|rcll|} \hline \dfrac{V}{A} &=& \dfrac{\frac43\pi r^3} {4\pi r^2} \\\\ \dfrac{V}{A} &=& \dfrac13\cdot \dfrac{4\pi}{4\pi}\cdot \dfrac{r^2}{r^2}\cdot r \\\\ \dfrac{V}{A} &=& \dfrac13\cdot r \\\\ V &=& \dfrac13 \cdot r \cdot A \\ \hline \end{array} \)
Radius r:
\(\begin{array}{|rcll|} \hline A &=& 4\pi r^2 \\ 4\pi r^2 &=& A \quad &| \quad : 4\pi \\ r^2 &=& \dfrac{A}{4\pi} \quad &| \quad \sqrt{} \\ r &=& \sqrt{ \dfrac{A}{4\pi} } \\ r &=& \dfrac12 \cdot \sqrt{ \dfrac{A}{\pi} } \\ \hline \end{array}\)
Volume V:
\(\begin{array}{|rcll|} \hline V &=& \dfrac13 \cdot r \cdot A \quad &| \quad r = \dfrac12 \cdot \sqrt{ \dfrac{A}{\pi} } \\\\ V &=& \dfrac13 \cdot \dfrac12 \cdot \sqrt{ \dfrac{A}{\pi} } \cdot A \\\\ \mathbf{V} &\mathbf{=}& \mathbf{ \dfrac16 \cdot \sqrt{ \dfrac{A^3}{\pi} } } \quad &| \quad A=75\ cm^2 \\\\ V &=& \dfrac16 \cdot \sqrt{ \dfrac{ ( 75\ cm^2)^3}{\pi} } \\\\ V &=& \dfrac16 \cdot \sqrt{ \dfrac{ 75^3}{\pi} }\ cm^3 \\\\ V &=& \dfrac16 \cdot \sqrt{ 134286.983234 }\ cm^3 \\\\ V &=& \dfrac16 \cdot 366.451883927 \ cm^3 \\\\ \mathbf{V} &\mathbf{=}& \mathbf{ 61.0753139879 \ cm^3 } \\ \hline \end{array}\)
The volume of the sphere is \(61.1\ cm^3\)
