Question

Yea I think and this the other day and

Answer 1

Given:

The radius of the tank is: R = 7.5 feet.

Volume rate is: dV/dt = 60 cubic feet per hour.

h = 2.8 feet.

To find:

V, R, h, dV/dt, dR/dt, dh/dt

Step-by-step explanation:

The given expression for volume is:

`V=\pi h^2(R-(1)/(3)h)`

Substituting the values in the above equation, we get:

`V=3.14*2.8^2*(7.5-(1)/(3)*2.8)=161.74\text{ cubic feet}`

The volume is 161.74 cubic feet.

The radius of the tank is given as: R = 7.5 feet

The height of the water in the tank is given as: h = 2.8 feet

The volume rate of water is given as: dV/dt = 60 cubic feet per hour

As the radius of the tank is constant, its derivative will be zero. Thus, dR/dt = 0.

dh/dt can be calculated by differentiating the given volume equation with respect to time as:

`\begin{gathered} (dV)/(dt)=\pi^2*(dh^2)/(dt)*(R-(1)/(3)(dh)/(dt)) \\ \\ (dV)/(dt)=\pi^2*(2dh)/(dt)*(R-(1)/(3)(dh)/(dt)) \\ \\ (dV)/(dt)=2\pi^2R*(dh)/(dt)-(2\pi^2)/(3)*((dh)/(dt))^2 \end{gathered}`

Substituting the values in the above equation, we get:

`\begin{gathered} 60=2\pi^2*7.5*(dh)/(dt)-(2\pi^2)/(3)*((dh)/(dt))^2 \\ \\ 60=148.044(dh)/(dt)-6.580((dh)/(dt))^2 \\ \\ 6.580((dh)/(dt))^2-148.044(dh)/(dt)+60=0 \end{gathered}`

Solving the above quadratic equation, we get:

dh/dt = 0.412 ft/hr or dh/dt = 22.086 ft/hr.

Final answer:

V = 161.74 cubic feet

R = 7.5 feet

h = 2.5 feet

dV/dt = 60 cubic feet per hour

dR/dt = 0

dh/dt = 22.086 ft/hr or 0.412 ft/hr