The measure of the diameter of a cylinder is (1.60± 0.01) cm and its length is (5.0± 0.1) cm. Calculate the percentage error in its volume.

The correct answer is 3.25%.

Step-by-Step Explanation:

To find the percentage error in the volume of the cylinder, we begin by writing the formula for the volume of a cylinder. The volume $V$ of a cylinder in terms of its diameter $d$ and length $l$ is given by:

$V$ = $\frac{\pi d^{2}l}{4}$

Since the constant factor $\frac{\pi }{4}$ has no error, the relative (fractional) error in the volume $\frac{\Delta V}{V}$ is determined by the errors in the measured quantities, diameter $d$ and length $l$. Using the rules of error propagation for powers and products, we get:

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

Let's identify the given values from the problem statement:
- Diameter of the cylinder, $d=1.60\text{cm}$
- Absolute error in diameter, $\Delta d=0.01\text{cm}$
- Length of the cylinder, $l=5.0\text{cm}$
- Absolute error in length, $\Delta l=0.1\text{cm}$

Now, we substitute these values into our error propagation equation:

$\frac{\Delta V}{V}=2\left(\frac{0.01}{1.60}\right)+\frac{0.1}{5.0}$

Let's calculate each term individually:

For the diameter term:
$2\times \frac{0.01}{1.60}=\frac{0.02}{1.60}=0.0125$

For the length term:
$\frac{0.1}{5.0}=0.02$

Adding the two fractional error terms together:

$\frac{\Delta V}{V}=0.0125+0.02=0.0325$

To find the percentage error in the volume, we multiply this fractional error by 100:

$\text{Percentage Error}=\frac{\Delta V}{V}\times 100\%=0.0325\times 100\%=3.25\%$

Thus, the percentage error in the volume of the cylinder is 3.25%.