M2 rate of change help plz!!
Water is pumped out at a rate of 3π cm3 per second from a cup that is in the shape of an inverted right circular cone. The height of the cup is 15 cm and its base radius is 5 cm. How fast is the water level falling when the depth of the water is 3 cm?
- 50418129Lv 77 年前最愛解答
Answer: 3 cm/s
let h cm and r cm be the depth and the base radius of water respectively
by similar triangle, 15 : 5 = h : r => r = h / 3
the volume of water = π r^2 h / 3 = π h^3 / 27
by differentiate both sides w.r.t. t,
d(the volume of water) / dt = (π h^2 / 9)(dh/dt)
- 3π = (π h^2 / 9)(dh/dt)
dh/dt = -27 / h^2
when the depth of the water is 3 cm, dh/dt = -27 / (3)^2 = -3
thus, the water level level is falling at a rate of 3 cm / second資料來源： knowledge