Question 300: Units , Dimensions and Error Analysis

Question

The dimensions of a cone are measured using a scale with a least count of $2 \mathrm{~mm}$. The diameter of the base and the height are both measured to be $20.0 \mathrm{~cm}$. The maximum percentage error in the determination of the volume is _______.

Accepted Answer

To determine the maximum percentage error in the volume of the cone, we first need to understand the dependence of the cone's volume on its dimensions: the diameter of the base and its height.
The volume $ V $ of a cone is given by the formula:
$$ V = \frac{1}{3} \pi r^2 h $$
where $ r $ is the radius of the base and $ h $ is the height.
Given that both the diameter ($ D $) of the base and the height ($ h $) are measured to be $ 20.0 \mathrm{~cm} $, we can find the radius $ r $ as follows:
$$ r = \frac{D}{2} = \frac{20.0 \mathrm{~cm}}{2} = 10.0 \mathrm{~cm} $$
Now, let's denote the errors in measuring the diameter and height as $ \Delta D $ and $ \Delta h $ respectively.
Since the measurements are taken with a scale having a least count of $ 2 \mathrm{~mm} $, we have:
$$ \Delta D = 2 \mathrm{~mm} = 0.2 \mathrm{~cm} $$
and
$$ \Delta h = 2 \mathrm{~mm} = 0.2 \mathrm{~cm} $$
To find the maximum percentage error in the volume, we need to use the formula for the propagation of relative errors. Considering the volume formula $ V = \frac{1}{3} \pi r^2 h $, the relative errors in $ r $ and $ h $ will propagate into the volume as follows:
The relative error in the radius $ \Delta r / r $ is:
$$ \left(\frac{\Delta r}{r}ight) = \left(\frac{\Delta D / 2}{D / 2}ight) = \left(\frac{\Delta D}{D}ight) = \frac{0.2 \mathrm{~cm}}{20.0 \mathrm{~cm}} = 0.01 $$
The relative error in the height $ \Delta h / h $ is:
$$ \left(\frac{\Delta h}{h}ight) = \frac{0.2 \mathrm{~cm}}{20.0 \mathrm{~cm}} = 0.01 $$
Since $ V $ is proportional to $ r^2 $ and $ h $, the total relative error in the volume is given by:
$$ \left(\frac{\Delta V}{V}ight) = 2 \left(\frac{\Delta r}{r}ight) + \left(\frac{\Delta h}{h}ight) $$
Substituting the relative errors, we get:
$$ \left(\frac{\Delta V}{V}ight) = 2 (0.01) + 0.01 = 0.02 + 0.01 = 0.03 $$
Hence, the maximum percentage error in the volume is:
$$ 0.03 	imes 100\% = 3\% $$
Therefore, the maximum percentage error in the determination of the volume is
3%
.

Units , Dimensions and Error Analysis

The dimensions of a cone are measured using a scale with a least count of 2 \mathrm{~mm}. The diameter of the base and the height are both measured to be 20.0 \mathrm{~cm}. The maximum percentage error in the determination of the volume is _______.

Integer / Numeric Answer Type