Question

Use the linear approximation to estimate the change in volume of a right-circular cone of radius r=40 cm if the height is increased from 40 to 41 cm. (Use decimal notation. Give your answer to three decimal places.)

Answer 1

Answer:

The change in volume is $1675.516cm^{3}$.

Step-by-step explanation:

The radius of the cone is 40 cm and the height increase from 40 to 41 cm.

$dh=41-40\\dh=1$

The volume of the cone is,

$v=\frac{1}{3}\pi r^{2}h$

Differentiate with respect to h,

$\dfrac{dv}{dh} =\dfrac{d}{dh}(\frac{1}{3}\pi r^{2}h)\\\dfrac{dv}{dh} =\dfrac{1}{3}\pi r^{2}\frac{dh}{dh}$

further simplify as:

$\dfrac{dv}{dh} =\dfrac{1}{3}\pi r^{2}\\dv =\dfrac{1}{3}\pi r^{2}dh$

substitute the value of 1 for  and 40 for r:

$dv =\dfrac{1}{3}\pi \times40^{2}\times1\\\\dv =\dfrac{1}{3}\pi \times1600\\\\dv =1675.516$

Hence, the change in volume is $1675.516cm^{3}$.