Grain is falling from a chute onto the ground, forming a conical pile whose diameter is always three times its height. How high is the pile (to the nearest hundredth of a foot) when it contains 1060 ft3 of grain?
Let $r$ be the radius of the base of the pile (in feet), and let $h$ be the height of the pile (in feet). Then the radius of the top of the pile is $3h$ (in feet). [asy]
size(180);
real r = 0.16657;
draw(Circle((0,0),r));
draw((0,0)--(0.85*r,0));
draw((0,0)--(0.85*r,0.85*r*3));
draw((0,0)--(-0.85*r,0.85*r*3));
draw((-0.85*r,0)--(-0.85*r,0.85*r*3),dashed);
label("$r$", (0.43*r,-0.15*r), UnFill);
label("$3h$", (-0.99*r,0.85*r), UnFill);
dot((0,0));
[/asy]
The area of the base of the pile is $\pi r^2$ (in square feet), and the area of the top of the pile is $\pi (3h)^2 = 9\pi h^2$ (in square feet). Because the grain forms a cone, the volume of the pile is \[\frac{1}{3}\cdot\pi r^2 h = \frac{1}{3}\cdot 9\pi h^2 h = 3\pi h^3.\]We are given that $3\pi h^3 = 1060$, so $h^3 = \frac{1060}{3\pi}$. Since $\pi \approx 3.14,$ $h^3 \approx \frac{1060}{3\cdot 3.14} = \frac{1060}{9.42} \approx 112.67.$
Therefore, $h\approx\boxed{4.93}$ (to the nearest hundredth of a foot).
To find the height of the pile when it contains 1060 ft³ of grain, we need to use the volume formula for a cone.
The volume of a cone is given by the formula: V = (1/3) * π * r² * h
Given that the diameter of the cone is always three times its height, we can deduce that the radius (r) is equal to (1/2) of the height (h).
Let's assume the height of the cone is 'h' feet. Therefore, the radius will be (1/2) * h = h/2.
Then, the volume of the cone can be written as: V = (1/3) * π * (h/2)² * h = (π/12) * h³
We are given that the volume (V) is 1060 ft³, so we can solve the equation for the height (h).
1060 = (π/12) * h³
To find the value of h, we can isolate it by dividing both sides of the equation by (π/12) and then taking the cube root.
1060 / (π/12) = h³
h³ = (1060 * 12) / π
h = (1060 * 12 / π)^(1/3)
Using a calculator, we can compute the approximate value of h to the nearest hundredth.
h ≈ 8.9 feet
Therefore, the height of the pile (to the nearest hundredth of a foot) when it contains 1060 ft³ of grain is approximately 8.9 feet.