Respuesta :
Answer:
[tex]21\pi[/tex] cubic meters per hour
Step-by-step explanation:
We know that the surface area of a sphere is,
[tex]A=4\pi r^2[/tex] ...........(1),
Where,
r = radius of the sphere,
We have, A = [tex]36\pi[/tex] square meters,
[tex]36\pi = 4\pi r^2[/tex]
[tex]9 = r^2[/tex]
[tex]\implies r = 3\text{ meters}[/tex]
Now, differentiating equation (1) with respect to t ( time ),
[tex]\frac{dA}{dt} = 8\pi r\frac{dr}{dt}[/tex]
We have,
[tex]\frac{dA}{dt}=14\pi\text{ square meters per hour}, r = 3\text{ meters}[/tex]
[tex]14\pi = 8\pi (3) \frac{dr}{dt}[/tex]
[tex]\frac{14}{24}=\frac{dr}{dt}[/tex]
[tex]\frac{7}{12} = \frac{dr}{dt}[/tex]
Also, the volume of a sphere is,
[tex]V = \frac{4}{3}\pi r^3[/tex]
Differentiating with respect to t(time)
[tex]\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}[/tex]
By substituting the values,
[tex]\frac{dV}{dt}=4\pi (3)^2 (\frac{7}{12})=36\pi \frac{7}{12}=21\pi\text{ cube meters per hour}[/tex]
Hence, the rate of change of the volume of the sphere at that instant would be [tex]21\pi[/tex] cubic meters per hour
