Answer:
Explanation:
To find how fast the height of the pile is increasing, we need to use the concepts of related rates. Let's break down the problem step by step.
First, we are given that sand is being dumped from a conveyor belt at a rate of 50 cubic feet per minute. This means that the volume of sand being dumped per minute is 50 cubic feet.
Next, we are told that the sand forms a conical pile with a base diameter twice smaller than its height. This implies that the base radius (r) of the pile is half the height (h).
Now, we need to determine the relationship between the height (h) and the volume of the cone (V). The volume of a cone is given by the formula V = (1/3)πr^2h, where π is a constant.
In our case, since the base radius (r) is half the height (h), we can substitute (h/2) for r in the volume formula:
V = (1/3)π(h/2)^2h
= (1/3)π(h^3/4)
= (π/12)h^3
We know that the volume is increasing at a rate of 50 cubic feet per minute, so we can differentiate the volume formula with respect to time (t) to find the rate of change of volume (dV/dt):
dV/dt = (d/dt)[(π/12)h^3]
= (π/4)h^2(dh/dt)
Now, we need to find the rate at which the height of the pile (h) is changing when the height is 8 feet. We can substitute h = 8 and dV/dt = 50 into the equation and solve for dh/dt:
50 = (π/4)(8^2)(dh/dt)
dh/dt = 50 / [(π/4)(8^2)]
≈ 0.796 feet per minute
Therefore, the height of the pile is increasing at a rate of approximately 0.796 feet per minute when the pile is 8 feet high.