Answer:
[tex]\dfrac{20\sqrt{5}}{3} \pi\; \sf cm^3[/tex]
Step-by-step explanation:
To find the volume of a sphere given its surface area is 20π cm², we can substitute the surface area into the surface area formula to find the radius, then substitute the radius into the volume formula.
The formula for the surface area of a sphere is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Surface Area of a Sphere}}\\\\SA=4\pi r^2\\\\\textsf{where:}\\ \phantom{ww}\bullet\;\textsf{$SA$ is the surface area.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius.}\end{array}}[/tex]
Substitute SA = 20π and solve for r:
[tex]4\pi r^2=20\pi\\\\4r^2=20\\\\r^2=5\\\\r=\sqrt{5}[/tex]
Therefore, the radius of the sphere is √5 cm.
The formula for the volume of a sphere is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Volume of a Sphere}}\\\\V=\dfrac{4}{3}\pi r^3\\\\\textsf{where:}\\ \phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius.}\end{array}}[/tex]
Substitute r = √5 into the volume formula:
[tex]V=\dfrac{4}{3} \pi \left(\sqrt{5}\right)^3\\\\\\\\V=\dfrac{4}{3} \pi \cdot 5\sqrt{5}\\\\\\\\V=\dfrac{4\cdot 5\sqrt{5}}{3} \pi\\\\\\\\V=\dfrac{20\sqrt{5}}{3} \pi\; \sf cm^3[/tex]
Therefore, the volume of the sphere with a surface area of 20π cm² is:
[tex]\Large\boxed{\boxed{\textsf{Volume}=\dfrac{20\sqrt{5}}{3} \pi\; \sf cm^3}}[/tex]
