Respuesta :
Answer:
[tex]9\pi \text{ cubic meters per hour}[/tex]
Step-by-step explanation:
Since, the surface area of a cylinder,
[tex]A= 2\pi r^2 + 2\pi rh[/tex] ................(1)
Where,
r = radius,
h = height,
If [tex]A= 36\pi\text{ square meters}, h = 3\text{ meters}[/tex]
[tex]36\pi = 2\pi r^2 + 2\pi r(3)[/tex]
[tex]18 = r^2 + 3r[/tex]
[tex]\implies r^2 + 3r - 18=0[/tex]
[tex]r^2 + 6r - 3r - 18 = 0[/tex] ( by middle term splitting )
[tex]r(r+6)-3(r+6)=0[/tex]
[tex](r-3)(r+6)=0[/tex]
By zero product property,
r = 3 or r = - 6 ( not possible )
Thus, radius, r = 3 meters,
Now, differentiating equation (1) with respect to t ( time ),
[tex]\frac{dA}{dt}= 4\pi r\frac{dr}{dt} +2\pi(r\frac{dh}{dt} + h\frac{dr}{dt})[/tex]
∵ h = constant, ⇒ dh/dt = 0,
[tex]\frac{dA}{dt} = 4\pi r \frac{dr}{dt} +2\pi h \frac{dr}{dt}[/tex]
We have, [tex]\frac{dA}{dt}=9\pi\text{ square meters per hour}, r = h = 3\text{ meters}[/tex]
[tex]9\pi = 4\pi (3) \frac{dr}{dt}+2\pi (3)\frac{dr}{dt}[/tex]
[tex]9\pi = (12\pi + 6\pi )\frac{dr}{dt}[/tex]
[tex]9\pi = 18\pi \frac{dr}{dt}[/tex]
[tex]\implies \frac{dr}{dt} =\frac{1}{2}\text{ meter per hour}[/tex]
Now,
Volume of a cylinder,
[tex]V=\pi r^2 h[/tex]
Differentiating w. r. t. t,
[tex]\frac{dV}{dt}=\pi ( r^2 \frac{dh}{dt}+h(2r)\frac{dr}{dt})=\pi ((3)(6) (\frac{1}{2})) = 9\pi \text{ cubic meters per hour}[/tex]
