[tex]\begin{gathered} _{_{_{_{_{}}}}}V=\frac{1}{3}\pi r^2h=\frac{\pi}{3}(r^2h) \\ \frac{dv}{dt}=\frac{\pi}{3}(2rh\frac{dr}{dt}+r^2\frac{dh}{dt} \\ \\ \text{Volume after 50 minutes =95}\times50=4.75m^3 \\ \\ \end{gathered}[/tex][tex]\begin{gathered} \frac{h}{r}=\frac{7}{5\text{ }} \\ h=\frac{7}{7}r \\ \\ \frac{dh}{dt}(\frac{7}{5}\frac{dr}{dt}) \\ \frac{dr}{dt}=\frac{5}{7}\frac{dh}{dr} \\ \\ \end{gathered}[/tex]
Substitute into the Volume
[tex]\begin{gathered} 4.75=\frac{1}{3}\pi r^2(\frac{7}{5}r) \\ \text{Simplify} \\ \\ 4.75=\frac{7}{15}\pi r^3 \\ \\ \text{make r subject the formula} \\ r^3=\frac{4.75\times15}{7\pi} \\ r=1.48m \\ h=2.07 \\ \\ 0.095=\frac{\pi}{3}(2(1.48)(2.07)\frac{5}{7}\frac{dh}{dt}+(1.48^2)\frac{dh}{dt}) \end{gathered}[/tex]
Make dh/dt the subject of the formula
[tex]\frac{dy}{dt}=0.0138143109\text{ m/min}[/tex]
