Sand falls from a conveyor belt at a rate of 20 m^3/min onto the top of a conical pile. The height of the pile is always 3/4 of the base diameter. Answer the following.
a.) How fast is the height changing when the pile is 10 m high?
Answer = m/min.
b.) How fast is the radius changing when the pile is 10 m high?
Answer = m/min.
To find the rate at which the height of the pile is changing when it is 10 m high (a), we can take the derivative of the volume of the cone with respect to time.
Let's start by finding the equation for the volume of the cone at any given time. We know that the height of the cone is always 3/4 of the base diameter.
Let's use h to represent the height of the cone, and r to represent the radius of the base of the cone. Since the height of the cone is 3/4 of the base diameter, we can write h = (3/4)r.
Now, let's express the volume of the cone, V, in terms of r and h. The formula for the volume of a cone is V = (1/3)πr^2h. Substituting h = (3/4)r, we get V = (1/3)πr^2(3/4)r = (3/12)πr^3 = (1/4)πr^3.
To find the rate at which the height is changing, dh/dt, when the pile is 10 m high, we need to find dh/dt when h = 10.
First, differentiate the volume equation with respect to time, t, using the chain rule:
dV/dt = (1/4)π(3r^2)(dr/dt).
Given that the sand falls from a conveyor belt at a rate of 20 m^3/min, the derivative of the volume with respect to time, dV/dt, is equal to -20.
Since we are looking for dh/dt when h = 10 and h = (3/4)r, we can substitute these values into our volume equation:
V = (1/4)πr^3, h = (3/4)r.
10 = (3/4)r, so r = (4/3)*10 = 40/3.
Now, substitute r = 40/3 into the equation dV/dt = (1/4)π(3r^2)(dr/dt):
-20 = (1/4)π(3(40/3)^2)(dr/dt).
Simplify the equation further:
-20 = (1/4)π(3(1600/9))(dr/dt).
-20 = (1/4)π(4800/9)(dr/dt).
Now, solve for dr/dt:
dr/dt = (-20)(4)(9)/(4800π).
Simplify the equation:
dr/dt = -1/(6π) m/min.
Therefore, when the pile is 10 m high, the radius is changing at a rate of -1/(6π) m/min.
To find the rate at which the radius is changing when the pile is 10 m high (b), we have already calculated that dr/dt = -1/(6π) m/min.