Question

A right circular cylinder has a radius of 40 cm and a height of 100 cm. Use differentials to estimate the change in volume of the cylinder if its height and radius are both increased by 1 cm. Give an exact answer.

Answer 1

Answer:

dV = 9600π = 30,171.4 cm³

Step-by-step explanation:

Volume of a cylinder = πr²h

V(r,h) = πr²h

Total Differential of V = dV

For a multi-variable function, the total differential is given as

dV = (∂V/∂r) Δr + (∂V/∂h) Δh

Δr = 1 cm

Δh = 1 cm

V(r,h) = πr²h

(∂V/∂r) = 2πrh

(∂V/∂h) = πr²

dV = (2πrh) (1) + (πr²) (1)

dV = (2πrh) + (πr²)

radius of 40 cm and a height of 100 cm

r = 40 cm

h = 100 cm

dV = (2πrh) + (πr²)

dV = (2π×40×100) + [π(40²)]

dV = 8000π + 1600π = 9600π

dV = 30,171.4 cm³

Hope this Helps!!!

Answer 2

The required answer is $\frac{dV}{dt}=9600\pi$

The volume of the cylinder:

The formula for the volume of the cylinder is,

$V=\pi r^2h$

It is given that,

$r=40cm\\h=100cm\\\frac{dh}{dt}=\frac{dr}{dt}=1 cm$

Now, the change in the volume can be given by,

$\frac{dV}{dt}=\pi r^2\frac{dh}{dt}+\pi \times 2r \times \frac{dr}{dt} \timesh\\\frac{dV}{dt}= \pi [(40)^2(1)+(2)(40)(1)(100)] \\\frac{dV}{dt}=9600\pi$

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