Answer:
Water level in the reservoir is falling at the rate of 0.398 ft per minute.
Step-by-step explanation:
From the figure attached,
Water level in the reservoir has been given as 10 feet and radius of the reservoir is 4 feet.
Let the level of water in the reservoir after time t is h where radius of the water level becomes r.
ΔABE and ΔCDE are similar.
Therefore, their corresponding sides will be in the same ratio.
[tex]\frac{r}{h}=\frac{4}{10}[/tex]
[tex]r=\frac{2}{5}h[/tex] --------(1)
Now volume of the water V = [tex]\frac{1}{3}\pi r^{2}h[/tex]
From equation (1),
V = [tex]\frac{1}{3}\pi (\frac{2}{5}h)^{2} h[/tex]
V = [tex]\frac{4\pi h^{2}\times h}{75}[/tex]
[tex]\frac{dV}{dt}=\frac{4\pi }{75}\times \frac{d}{dt}(h^{3})[/tex]
[tex]\frac{dV}{dt}=\frac{4\pi }{75}\times (3h^{2})\frac{dh}{dt}[/tex]
[tex]\frac{dV}{dt}=\frac{12\pi h^{2}}{75}\times \frac{dh}{dt}[/tex]
Since [tex]\frac{dV}{dt}=5[/tex] feet³ per minute and h = 5 feet
[tex]5=\frac{12\pi (5)^{2}}{75}\times \frac{dh}{dt}[/tex]
[tex]5=4\pi \frac{dh}{dt}[/tex]
[tex]\frac{dh}{dt}=\frac{5}{4\pi}[/tex]
[tex]\frac{dh}{dt}=0.398[/tex] feet per minute
Therefore, water level in the reservoir is falling at the rate of 0.398 feet per minute.
