Sand is being dropped at the rate 10 metercube/min onto a conical pile. If the height of the pile is always twice the base radius, at what rate is the increasing when the pile is 8 m high?
v = 1/3 pi r^2 (2r) = 2/3 pi r^3
at r=8,
dv/dt = 2 pi r^2 dr/dt
Now we just have to decide what "at what rate is the increasing" means
To find the rate at which the height of the pile is increasing when it's 8 meters high, we need to use related rates.
Let's define some variables:
h = height of the pile (in meters)
r = base radius of the pile (in meters)
V = volume of the sand cone (in cubic meters)
t = time (in minutes)
We are given two pieces of information:
1. The height of the pile is always twice the base radius:
h = 2r
2. Sand is being dropped at a rate of 10 cubic meters per minute:
dV/dt = 10
We want to find dh/dt, the rate at which the height of the pile is increasing when it's 8 meters high.
To do this, we need to express dh/dt in terms of known variables. We can relate the variables using the formula for the volume of a cone:
V = (1/3) * π * r^2 * h
Now, we can differentiate both sides of the equation with respect to time (t):
dV/dt = (1/3) * π * (2r * dr/dt * h + r^2 * dh/dt)
Using the chain rule for differentiation.
Substituting the known values, we have:
10 = (1/3) * π * (2r * dr/dt * (2r) + r^2 * dh/dt)
Simplifying further:
10 = (4/3) * π * r^2 * dr/dt + (2/3) * π * r^3 * dh/dt
We need to find dh/dt when h = 8 meters. From the given information, we know h = 2r. So, when h = 8 meters, r = h/2 = 4 meters.
Plugging in the known values:
10 = (4/3) * π * (4^2) * dr/dt + (2/3) * π * (4^3) * dh/dt
10 = (4/3) * π * 16 * dr/dt + (2/3) * π * 64 * dh/dt
Plugging in the value of dh/dt and solving for dr/dt:
dh/dt = (10 - (4/3) * π * 16 * dr/dt) / ((2/3) * π * 64)
Now we can solve for dr/dt by substituting dh/dt = 10 and h = 8:
10 = (10 - (4/3) * π * 16 * dr/dt) / ((2/3) * π * 64)
10 * (2/3) * π * 64 = 10 - (4/3) * π * 16 * dr/dt
(2/3) * π * 64 * dr/dt = 10 - (10 * (2/3) * π * 64)
(2/3) * π * 64 * dr/dt = 10 * (1 - (2/3) * π * 64)
dr/dt = 10 * (1 - (2/3) * π * 64) / ((2/3) * π * 64)
Calculating the final numerical value will give us the rate at which the base radius is changing with respect to time when the pile is 8 meters high.