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

The correct answer is 3.

Step 1: Understand the formula for the volume of a cone
The volume $V$ of a cone with radius $r$ and height $h$ is given by the formula:

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

Since the diameter $d=2r$, we can rewrite the radius in terms of diameter as $r=\frac{d}{2}$.
Substituting $r$ into the volume formula:

$V=\frac{1}{3}\pi {\left(\frac{d}{2}\right)}^{2}h=\frac{1}{12}\pi d^{2}h$

Step 2: Formulate the fractional error in volume
Taking the logarithm and differentiating, or using the standard rule for propagation of relative errors, the fractional (relative) maximum error in volume $\frac{\Delta V}{V}$ is:

$\frac{\Delta V}{V}=2\frac{\Delta d}{d}+\frac{\Delta h}{h}$

Hence, the maximum percentage error in the determination of volume is:

$\text{Percentage Error in}V=\left(2\frac{\Delta d}{d}+\frac{\Delta h}{h}\right)\times 100\%$

Step 3: Extract the given measurements and absolute error
Least count of the measuring scale = $2\text{mm}=0.2\text{cm}$.
Therefore, the absolute error in measuring both the diameter and height is $\Delta d=\Delta h=0.2\text{cm}$.
Given measured values:
Diameter, $d=20.0\text{cm}$
Height, $h=20.0\text{cm}$

Step 4: Calculate the percentage error
Substitute the values into the error formula:

$\frac{\Delta d}{d}=\frac{0.2}{20.0}=\frac{2}{200}=0.01$

$\frac{\Delta h}{h}=\frac{0.2}{20.0}=\frac{2}{200}=0.01$

Now, calculate the maximum fractional error in volume:

$\frac{\Delta V}{V}=2\left(0.01\right)+0.01=0.02+0.01=0.03$

Multiply by 100 to get the percentage error:

$\text{Percentage Error in}V=0.03\times 100\%=3\%$

Thus, the maximum percentage error in the volume determination is 3.