Question

The volume of a circular cylinder is calculated using the formula = ^2*2ℎ. If the height ℎ is known to be equal to the diameter 2 and ℎ is measured at a percentage error of 2%, find the percentage error in calculating the volume of the cylinder.

Answer 1

Answer:

6%

Step-by-step explanation:

From the information given:

The height h is said to be equal to the diameter which is 2r

h = 2r

r = h/2

Recall that: The volume for calculating a circular cylinder is:

$V = \pi r^2 h$

$V = \pi (\dfrac{h}{2})^2 h$

$V =\dfrac{\pi h^3}{2}$

$\dfrac{dV}{dh} = \dfrac{3 \pi h^3}{4}$

Thus, the percentage error of the height can now be calculated as:

$\dfrac{dh }{h} \times 100 = 2$

$dh =\dfrac{h}{50}$

Now taking the differential of the volume, we have:

$dV = \dfrac{dV}{dh}* dh$

$dV = \dfrac{3 \pi h^2}{4}* \dfrac{h}{50}$

FInally, the %age change in the volume is calculated as follows:

$\dfrac{dV}{V} = \dfrac{ \dfrac{3 \pi h^2}{4}* \dfrac{h}{50}}{\dfrac{\pi h^3}{2}}$

$\dfrac{dV}{V} = \dfrac{3}{50} \times 100 \%$

$\dfrac{dV}{V} =6 \%$

Thus;  the percentage error in calculating the volume of the cylinder is 6%