Question

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?

Answer 1

Answer:

(a)$\frac{dV}{dt}= 2\pi r h \frac{dr}{dt}$

(b)$\frac{dV}{dt}= \pi r^2 \frac{dh}{dt}$

(c) $\frac{dV}{dt} = \pi r^2\frac{dh}{dt}+2\pi r h\frac{dr}{dt}$

Explanation:

Differentiating Rules:

1. $\frac{dx^n}{dx}= nx^{n-1}$
2. $\frac{dx}{dx}=1$
3. $\frac{d}{dx}(mn)= m\frac{dn}{dx}+n\frac{dm}{dx}$  [ m and n are the function of x]
4. $\frac{d}{dx}(cn)=c \frac{dn}{dx}$ [ here c is constant and n is function of x]

Given that,

$V= \pi r^2h$

(a)

$V= \pi r^2h$

Differentiating with respect to t

$\frac{dV}{dt}= \frac{d}{dt}(\pi r^2h)$

$\Rightarrow \frac{dV}{dt}= \pi h \frac{d}{dt}(r^2)$    [ here $\pi h$ is constant]

$\Rightarrow \frac{dV}{dt}= \pi h 2r \frac{dr}{dt}$

$\Rightarrow \frac{dV}{dt}= 2\pi r h \frac{dr}{dt}$

(b)

$V= \pi r^2h$

Differentiating with respect to t

$\frac{dV}{dt}= \frac{d}{dt}(\pi r^2h)$

$\Rightarrow \frac{dV}{dt}= \pi r^2 \frac{dh}{dt}$

(c)

$V= \pi r^2h$

Differentiating with respect to t

$\frac{dV}{dt}= \frac{d}{dt}(\pi r^2h)$

$\Rightarrow \frac{dV}{dt} = \pi r^2\frac{dh}{dt}+\pi h\frac{d}{dt}(r^2)$

$\Rightarrow \frac{dV}{dt} = \pi r^2\frac{dh}{dt}+2\pi r h\frac{dr}{dt}$

Answer 2

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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