Answer:
The rate of the height of the pile changing when the pile is 12 feet high is 101.828.
Step-by-step explanation:
Sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute.
Therefore, if the volume is denoted by ''V" then the rate of change of the volume is 10.
So,
[tex]\bold{\dfrac{dV}{dt}=10}[/tex]
The diameter of the base of the cone is approximately three times the altitude.
Thus,
[tex]D=3h\\2r=3h\\r=\dfrac{3h}{2}[/tex]
Now,
The expression for finding the volume of the cone is formulated as:
[tex]\begin{aligned} V=\dfrac{1}{3} \pi r^2h\\V=\dfrac{1}{3} \pi (\dfrac{3h}{2})^2h\\V=\dfrac{3\pi h^3}{4}\end{aligned}[/tex]
Now,
Differentiate the above equation with respect to time.
[tex]\dfrac{dV}{dt}=\dfrac{d}{dt}\left[\dfrac{3\pi h^3}{4}\right]\\\dfrac{dV}{dt}=\left[\dfrac{9\pi h^2}{4}\right]\dfrac{dh}{dt}[/tex]
Now,
Rate is the height of the pile changing is [tex]\dfrac{dh}{dt}[/tex].
Height is given in the question that is 12 feet.
Therefore,
[tex]\dfrac{dV}{dt}=\left[\dfrac{9\pi (12)^2}{4}\right]\dfrac{dh}{dt}\\10=324 \pi \dfrac{dh}{dt}\\\dfrac{dh}{dt}=32.4 \pi\\\dfrac{dh}{dt}=101.828[/tex]
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