The height of the pile is increasing 8/(pi*125) ft/min when the height is 25 ft.
In the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations, calculus, sometimes known as infinitesimal calculus or "the calculus of infinitesimals," is the study of continuous change in mathematics.
We are given dV/dt=10 cubic ft/min and we want to find dh/dt. Therefore, first we need to write a formula relating v and h.
V=pi*r^2*h/3 since d=h at any moment, we know that r=h/2 at any moment and if we substitute h/2 for r, we get
V=pi(h/2)^2*h/3 and if we simplify this, we get
V=pi*(h^3)/12
Now we need to take the derivative of both sides with respect to t
dV/dt=pi*(3*h^2)*/12 *dh/dt=pi*h^2/4*dh/dt, substitute 10 for dV/dt and 25 for h and solve for dh/dt and we get
10=pi^625/4*dh/dt ==> dh/dt=10*4/(625*pi)=8/(125*pi) ft/min,
that is, the height of the pile is increasing 8/(pi*125) ft/min when the height is 25 ft.
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