The volume $V$ of a right circular cylinder is given by:

$V = \pi r^2 h$

Where:

• $r$ is the radius of the base.
• $h$ is the height.

If there are relative errors of 1% in $r$ and 2% in $h$, we use error propagation to find the percentage error in the volume.

Step 1: General formula for relative error in $V$

Taking the derivative of $V = \pi r^2 h$, the relative error is:

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

Here:

• $\frac{\Delta r}{r}$ = relative error in radius ($r$).
• $\frac{\Delta h}{h}$ = relative error in height ($h$).

Step 2: Substitute the given errors

• $\frac{\Delta r}{r} = 1\% = 0.01$
• $\frac{\Delta h}{h} = 2\% = 0.02$

Substitute into the formula:

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

$\frac{\Delta V}{V} = 0.02 + 0.02 = 0.04$

Step 3: Convert to percentage

$\frac{\Delta V}{V} \times 100 = 0.04 \times 100 = 4\%$

Final Answer:

The error in the calculated volume is 4%.

Would you like a step-by-step explanation of error propagation? Here are 5 related questions to deepen your understanding:

1. What is the importance of error propagation in measurements?
2. How does squaring the radius affect the error in the volume formula?
3. How would the error change if the error in radius was larger than the error in height?
4. Can you derive the error formula if the volume depends on more variables, like a cone or sphere?
5. How does relative error differ from absolute error in practice?

Tip: When working with formulas, always account for how the powers (like $r^2$) amplify the errors in measurements.