a conveyor belt pours gravel into a pile at a rate of 20 lbs per minue. the pile is shaped like an upside-down cone. the proportions of the cone are such that when the pile is complete, the diameter of the base is 6 ft and the height is 10 feet. how fast is the height of the pile changing when the pile is 7 ft high?
I keep getting stuck!
let the radius be r and the height be h
Assuming the cone maintains the same shape
2πr=6
r = 3/π
and r/h = (3/π)/10
r = 3h/(10π)
V = (1/3)πr^2 h
= (1/3)π(9h^2/(100π^2))h
= 3h^3/(100π)
dV/dt = (9/100π)h^2 dh/dt
At this point you will have to know what the relationship between lbs of concrete and cubic feet is, I work in metric and no longer know those units.
To find the rate at which the height of the pile is changing, we can use related rates and the properties of similar cones.
Let's denote the height of the pile as 'h' and the diameter of the base as 'd'. From the problem, we know that the diameter of the base is 6 ft when the pile is complete. Therefore, the radius of the base is half of the diameter, which is 6 ft / 2 = 3 ft.
We are given that the pile is shaped like an upside-down cone, so we can use the formula for the volume of a cone:
V = (1/3) * π * r^2 * h,
where V is the volume, π is approximately 3.14159, r is the radius of the base, and h is the height of the pile.
Since the pile is being poured at a constant rate of 20 lbs per minute, we can also express the volume of the pile as the rate at which gravel is being added:
dV/dt = 20 lbs/min.
To find how fast the height of the pile is changing (dh/dt) when the pile is 7 ft high, we need to find dh/dt.
To relate the variables, we note that the diameter of the pile is directly proportional to the height. So, we can use similar triangles to establish the proportionality:
d/h = 6 ft / 10 ft.
Now, let's differentiate both sides of the equation with respect to time (t):
(d/h) * (dh/dt) = (6/10) * (dd/dt).
Since we want to find dh/dt, we can solve for it:
dh/dt = (6/10) * (dd/dt) * (h/d).
To evaluate each term in the equation, we already have h = 7 ft and d = 6 ft, and we know that dd/dt is the rate at which the diameter is changing. However, we don't have the value of dd/dt.
To find dd/dt, we need to relate it to the rate at which the volume is changing (dV/dt). For this, we can differentiate the volume equation, V = (1/3) * π * r^2 * h, with respect to time:
dV/dt = (1/3) * π * (2r * dr/dt) * h + (1/3) * π * r^2 * dh/dt,
since the derivative of r^2 with respect to time is 2r * dr/dt.
Now, substituting the given values:
20 lbs/min = (1/3) * π * (2 * 3 ft * dr/dt) * 7 ft + (1/3) * π * (3 ft)^2 * dh/dt.
Simplifying the equation:
20 lbs/min = (2/3) * π * 21 ft * dr/dt + (1/3) * π * 9 ft^2 * dh/dt.
Now, we need to find dr/dt, the rate at which the radius is changing. Since we don't have that information, it seems there is a missing piece of information in the problem statement.
If you have additional data or constraints regarding the rate at which the radius is changing, please provide it so that we can assist you further in solving the problem.