ANSWERS: 1

This is a related rates problem. The key is to find a relation (read, "equation") between the volume V and the height of the pile, h. Since we're trying to find how fast the height is increasing, we're looking for h', the first timederivative of h. Since the pile is a right, circular cone whose base diameter is equal to the height, we can write the volume of the cone as follows: V = (1/3)*(pi)*r^2*h = (pi/3)*(h/2)^2*h = (pi/12)*h^3 since the the diameter D = 2r = h. Since we're looking for h', and we're given h=24ft and V'=20ft^3/min, a good idea would be to implicitly differentiate this equation since it'll yield a differential equation with everything we need to solve for h'. So we implicitly differentiate to get V' = (pi/12)*(3*h^2*h') = (pi/4)*h^2*h' Which means that we can solve for h': h' = 4*V'/(pi*h^2) Now we just plug in V'=20 and h=24 to get h' = 4*(20)/(pi*(24)^2) = 5/(36*pi) which is about equal to .0442097 ft/min. This makes sense because it's positive, which means the pile is getting bigger, which we would expect since V' is positive.
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