Gravel is being dumped from a conveyor belt at a rate of 50 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal to each other. How fast is the height of the pile increasing when the pile is 12 feet high?
Hint: The volume of a right circular cone with height and radius of the base is given by V=(1/3)(pi)(r^2)(h).
V=(1/3)(pi)(r^2)(h).
But we are told that 2r=h or r = h/2
then
V=(1/3)(pi)(h^2/4)(h).
=(1/12)pi(h^3)
dV/dh = (1/4)pi(h^2)dh/dt
when h = 12, and dV/dt = 50
50 = (1/4)pi(144)dh/dt
dh/dt = 200/(144pi) = 25pi/18 ft/min
this is wrong
To find the rate at which the height of the pile is changing, we need to use the formula for the volume of a right circular cone:
V = (1/3)(π)(r^2)(h)
where V is the volume, π is a mathematical constant (approximately 3.14159), r is the radius of the base, and h is the height of the cone.
In this case, we know that the rate at which gravel is being dumped is 50 cubic feet per minute. This gives us:
dV/dt = 50
To find the rate at which the height of the pile is changing when it is 12 feet high, we need to find dh/dt.
Given that the base diameter and the height of the pile are always equal, the radius of the base is half the base diameter. Let's call the base diameter D, so the radius r = D/2.
We also know that the height of the cone is equal to the radius, so h = r.
To relate the height and radius of the cone, we can use similar triangles. Let's call x the distance from the center of the base to the vertex of the cone.
Using similar triangles, we have: x/h = (D/2)/h
Simplifying, we get x = D/2
Now let's differentiate the equation V = (1/3)(π)(r^2)(h) with respect to time t:
dV/dt = (1/3)(π)[2r(dr/dt)(h) + r^2(dh/dt)]
Given that dV/dt = 50 and r = D/2, we can substitute these values:
50 = (1/3)(π)[2(D/2)(dD/dt)(D/2) + (D/2)^2(dh/dt)]
Simplifying further:
50 = (1/3)(π)[(D^2/4)(dD/dt) + (D^2/4)(dh/dt)]
Simplifying the expression inside the brackets:
50 = (1/3)(π)(D^2/4)(dD/dt + dh/dt)
Rearranging the equation to solve for dh/dt:
dh/dt = [50(3)/(π)(D^2/4)] - (dD/dt)
Now, when the pile is 12 feet high, the radius D can be calculated by finding the diameter of the base of the cone, which is twice the radius:
D = 2r = 2(12) = 24 feet
So we have D = 24 feet.
Substituting this value into the equation for dh/dt:
dh/dt = [50(3)/(π)(24^2/4)] - (dD/dt)
Simplifying:
dh/dt = (150/π)(12^2) - (dD/dt)
Calculating (150/π)(12^2) ≈ 908.86:
dh/dt = 908.86 - (dD/dt)
To find the value of dh/dt, we need to know the rate at which the base diameter dD/dt is changing. Since it is not given in the problem statement, we cannot determine the value of dh/dt without further information.