The volume V of a right circular cylinder of radius r and height h is V=πr2h. (a) How is dVdt related to drdt if h is constant and r varies with time? (Enter drdt as dr/dt.) dVdt= (dh)/(dt) (b) How is dVdt related to dhdt if r is constant and h varies with time? (Enter dhdt as dh/dt.) dVdt= (c) How is dVdt related to dhdt and drdt if both h and r vary with time?

A) dV/dt is related to dr/dt when h is constant and r varies with time as;

dV/dt = 2πrh(dr/dt)

B) dV/dt is related to dh/dt when r is constant and h varies with time as; dV/dt = πr²(dh/dt)

C) dV/dt is related to dr/dt and dh/dt when both h and r vary with time as;

dV/dt = πr²(dh/dt) + 2πrh(dr/dt)

A) We are given the volume of a right circular cylinder as;

V = πr²h

If h is constant and r varies with time, then we are differentiating only r with respect to t. Thus by partial differentiation, we have;

dV/dt =  πh(d(r²)/dt)

dV/dt = 2πrh(dr/dt)

B) Now, we are told that r is constant and h varies with time. Thus;

dV/dt = πr²(d(h)/dt)

Differentiating h with respect to t gives;

dV/dt = πr²(dh/dt)

C) Now, both h and r vary with time. Thus, differentiating both r and h with respect to t by  partial differentiation gives;

dV/dt = πr²(d(h)/dt) + πh(d(r²)/dt)

⇒ dV/dt = πr²(dh/dt) + 2πrh(dr/dt)

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(a)

(b)

(c)

Explanation:

Differentiating Rules:

Given that,

(a)

Differentiating with respect to t

   [ here is constant]

(b)

Differentiating with respect to t

(c)

Differentiating with respect to t

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