Question

A water filter has the shape of an inverted right circular cone with base radius 3 meters and height 9 meters. water is being pumped into the filter at the rate of 10 meters3/sec. find the rate, in meters/sec, at which the water level is rising when the water is 3 meters deep. give 2 decimal places for your answer. type your answer in the space below. if your answer is a number less than 1, place a leading "0" before the decimal point (ex: 0.35).

Answer 1

Hello there.

A water filter has the shape of an inverted right circular cone with base radius 3 meters and height 9 meters. water is being pumped into the filter at the rate of 10 meters3/sec. find the rate, in meters/sec, at which the water level is rising when the water is 3 meters deep. give 2 decimal places for your answer.

9.55 m/s

Answer 2

The volume of a cone is
V = (1/3)πr³
We are given with radius and the height of cone, so we can solve for the radius as function of water level using ratio and proportion
3/9 = r/h
r = h/3
Substituting
V = πh³/81
Taking the derivative
dV/dt = πh²/27 dh/dt
Solving for dh/dt given
dV/dt = 10 m³/s
and
h = 3 m
dh/dt = 9.55 m/s