Start with Volume equation for a Cone in terms of base Area.
\(V = \frac{1}{3} A h\)
Next relate height in terms of base Area: (Note base = pi*r^2)
\(h = 2D = 4r = 4 \sqrt {\frac{A}{\pi}}\)
New volume equation is:
\(V = \frac{4}{3 \sqrt{\pi}} A^{3/2}\)
Take derivative with respect to time:
\(\frac{dV}{dt} = (\frac{4}{3\sqrt{\pi}})(\frac{3}{2}) \sqrt{A} \frac{dA}{dt}\)
Sub in rate for volume, solve for dA/dt
\(\frac{dA}{dt} = \frac{10 \sqrt{\pi}}{\sqrt{A}}\)
Finally we need the Area after 5 min, given the volume after 5 min is 100.
Go back to Volume equation and solve for sqrt(A)
\(100 = \frac{4}{3 \sqrt{\pi}} (\sqrt{A})^3 \\ \sqrt{A} = (75 \sqrt{\pi})^{1/3}\)
Final Answer:
\(\frac{dA}{dt} = \frac{10 \sqrt{\pi}}{(75 \sqrt{\pi})^{1/3}} = 3.47\)
