Answer:
[tex]V'=-1.3824\ cm^3/s[/tex]
Step-by-step explanation:
Rate of Change
The surface area of a sphere of ratio r is
[tex]\displaystyle A=4\pi r^2[/tex]
And its volume is
[tex]\displaystyle V=\frac{4}{3}\pi r^3[/tex]
We know the surface area is [tex]24\ cm^2[/tex], let's find the ratio
[tex]\displaystyle r=\sqrt{\frac{A}{4\pi }}=\sqrt{\frac{24}{4\pi }}=1.382\ cm[/tex]
We'll find the rate of change of the surface area with respect to the time by taking the derivative
[tex]\displaystyle A'=8\pi r r'[/tex]
Solving for r'
[tex]\displaystyle r'=\frac{A'}{8\pi r }[/tex]
Since
[tex]A'=-2 \ cm^2/s[/tex]
(note the correction of the units)
[tex]\displaystyle r'=\frac{-2}{8\pi r }=-0.0576\ cm/s[/tex]
The change of the volume is obtained by taking the derivative:
[tex]V'=4\pi r^2 r'=4\pi \cdot 1.382^2 \cdot (-0.0576)[/tex]
[tex]\boxed{V'=-1.3824\ cm^3/s}[/tex]
