sand is being poured onto the ground forming a conical pile whose height equals 1/4 of the diameter of the base. The sand is falling at a rate of 20cm^3/sec. How fast is the height of the sand pile increasing when it is 3cm high?
To find how fast the height of the sand pile is increasing when it is 3 cm high, we can use related rates and differentiation.
Let's denote:
- h as the height of the conical sand pile,
- r as the radius of the base of the conical sand pile,
- V as the volume of the sand pile.
According to the problem, the height of the sand pile (h) is equal to 1/4 of the diameter of the base. Since the diameter is twice the radius, we have h = 1/4 * 2r = r/2.
The volume (V) of a cone can be calculated using the formula V = (1/3) * π * r^2 * h.
We are given that the sand is falling at a rate of 20 cm^3/sec, which is the rate of change of volume with respect to time (dV/dt). We need to find the rate of change of the height (dh/dt) when h = 3 cm.
Now, let's differentiate the volume equation with respect to time (t):
dV/dt = (1/3) * π * 2r * dr/dt * h + (1/3) * π * r^2 * dh/dt.
Since the height (h) and radius (r) are related, we can substitute r = 2h into the equation:
dV/dt = (1/3) * π * 2(2h) * dr/dt * h + (1/3) * π * (2h)^2 * dh/dt
=> 20 = (8/3) * π * h * dr/dt + (4/3) * π * h^2 * dh/dt.
We know that dr/dt represents the rate at which r is changing with respect to time. However, in this problem, the radius is not changing since sand is being poured onto the ground. Therefore, dr/dt is equal to 0.
Substituting 0 for dr/dt, we can simplify the equation:
20 = (8/3) * π * h * 0 + (4/3) * π * h^2 * dh/dt
=> 20 = 0 + (4/3) * π * h^2 * dh/dt.
Now, let's solve for dh/dt by isolating it:
(4/3) * π * h^2 * dh/dt = 20
=> dh/dt = 20 / [(4/3) * π * h^2]
=> dh/dt = 15 / (π * h^2).
Now we can substitute h = 3 cm into the equation to find the rate at which the height is increasing when h = 3 cm:
dh/dt = 15 / (π * (3)^2)
=> dh/dt = 15 / (9π)
=> dh/dt ≈ 0.530 cm/sec.
Therefore, when the height of the sand pile is 3 cm, the height is increasing at a rate of approximately 0.530 cm/sec.