Respuesta :
We know that the volume of a sphere of radius r is given by
[tex]V=\frac{4}{3}\pi r^3[/tex]
Now, we have been given the initial volume was 4,188.79 cubic centimeters. Let us find the radius at this time.
Substituting the value of volume in the above equation, we get
[tex]4,188.79=\frac{4}{3}\pi r^3\\ \\ 4\pi r^3=4,188.79 \times 3\\ \\ r^3=\frac{4188.79 \times 3}{4\pi}\\ \\ r^3=1000\\ \\ r=10[/tex]
Now, the final volume is given by 14,137.167 cubic centimeters. Thus, we have
[tex]14137.167 =\frac{4}{3}\pi r^3\\ \\ 4\pi r^3=14137.167 \times 3\\ \\ r^3=\frac{14137.167\times 3}{4\pi}\\ \\ r^3=3375\\ \\ r=15[/tex]
Now, let us find the surface areas for these two radii.
For [tex]r=10[/tex]
Surface area is given by
[tex]S=4\pi r^2\\ \\ S=4\times 3.14 \times (10)^2\\ \\ S=1256 \text{ square centimeters}[/tex]
Similarly, for
[tex]r=15\\ \\ S=4\times 3.14\times (15)^2\\ \\ S=2826 \text{ square centimeters}[/tex]
Therefore, the increase in the surface area is given by
[tex]\Delta S= 2826-1256= 1570 \text{ square centimeters}[/tex]
Hence, the average rate at which the surface area is changing is given by
[tex]\frac{1570}{12}=130.8[/tex]
Therefore, the average rate at which the surface area is changing is given by 130.8 square centimeters per second.
