Related rates

Question

A coffee filter has the shape of an inverted cone. Water drains out of the filter at a rate of 10cm3 per minute. When the depth of the water in the cone is 8cm, the depth is decreasing at a rate of 2cm per minute. What is the ratio of the height of the cone to its radius?

Answer 1

Volume v = πr²h; dv/dt = π(2rhdr/dt + r²dh/dt); let h/r = q; h = qr; r = h/q, dr/dt = dhr/dt/q;

dv/dt = π(2h²/q²dh/dt + h²/q²dh/dt); dv/dt = 3πh²/q²dh/dt; q = (3πh²dh/dt/(dv/dt))^1/2 = ( 3π·64cm²·

(- 2cm/s)/(-10cm³/s))^1/2 = 10.98