I'm currently taking MAT 265 in Arizona State online and came across this problem:
Water is leaking out of an inverted conical tank at a rate of $6700.0$ cubic centimeters per minute at the same time that water is being pumped into the tank at a constant rate. The tank has height $14.0$ meters and the diameter at the top is $5.0$ meters. If the water level is rising at a rate of $16.0$ centimeters per minute when the height of the water is $3.0$ meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.
My thought process is like so:
- Change in volume can be described in terms of leaking and pumping of the water
$$\frac{dV}{dt}=R-0.0067\frac{m^3}{min}$$
- It can also be described in terms of the conic volume formula:
$$\frac{dV}{dt}=\frac{1}{3}\cdot \pi \cdot \left(2\cdot r\cdot \frac{dr}{dt}\cdot h+r^2\cdot \frac{dh}{dt}\right)$$
We don't know $\frac{dr}{dt}$, however we can use the chain rule and rewrite it as $\frac{dr}{dh}\cdot\frac{dh}{dt}$. Now, if we imagine an inverted conic, as we go up from the tip to the base at constant speed, radius also grows linearly, so $\frac{dr}{dh}$ is the average change over the whole height, so
$$\frac{dr}{dh}=\frac{5-0}{14-0}=\frac{5}{14}$$ Also, $r$ in this case becomes: $$r=h\cdot\frac{dr}{dh}=3\cdot\frac{5}{14}=\frac{15}{14}m$$ 3. With that information we simplify, equate two definitions of $\frac{dV}{dt}$ and solve for R. $$\frac{dV}{dt}=\frac{1}{3}\cdot \pi \cdot \left(2\cdot r\cdot \frac{dr}{dh}\cdot \frac{dh}{dt}\cdot h+r^2\cdot \frac{dh}{dt}\right)$$ $$R-0.0067=\frac{1}{3}\cdot\pi\cdot\left(2\cdot r\cdot\frac{dr}{dh}\cdot\frac{dh}{dt}\cdot h+r^2\cdot\frac{dh}{dt}\right)$$ $$R=\frac{1}{3}\cdot\pi\cdot\left(2\cdot r\cdot\frac{dr}{dh}\cdot\frac{dh}{dt}\cdot h+r^2\cdot\frac{dh}{dt}\right)+0.0067$$ $$R=\frac{1}{3}\cdot\pi\cdot\left(2\cdot \frac{15}{14}\cdot\frac{5}{14}\cdot0.16\cdot 3+\left(\frac{15}{14}\right)^2\cdot0.16\right)+0.0067$$ $$R=0.58372722\frac{m^3}{min}=583727.22\frac{cm^3}{min}$$
However, this doesn't seem like the correct answer and there's certainly something wrong here. What am I missing and what would be the correct solution?