A differentiation problem I was attempting is:
"Water is dripping from a conical funnel at a uniform rate of 4 cm$^3 /s$ through a tiny hole at the vertex in the bottom. When the slant height of the water is 3 cm, find the rate of decrease of slant height of the water cone. (Given: vertical angle of funnel =120 degree)."
My approach :
I took the formula for volume of a cone and wrote radius, r and height h as: $ r = \sqrt3l/2 $ and $ h = l/2$, using trigonometric formula for sine and cosine. Finally, I got $dl/dt = 32/(27\pi) cm/s$.
But the solution given is: $dl/dt = 8/(9\pi) cm/s$. In the answer, instead of taking r as $(\sqrt3l)/2$ they put the value of l here and so put $r = (3\sqrt3)/2$. Also, for the height, used this formula: $h^2 = l^2 - r^2 = l^2 - (27/4) \implies h = \sqrt{l^2- 27/4} $.
This is what I don't understand: How can we put a value for radius when it too is changing with time just like the slant length and height? And why is my solution incorrect?