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

Answer

To calculate the percentage error in the volume of a cylinder given the diameter and length with their respective uncertainties, we can follow these steps:

### Step 1: Identify the given values
- Diameter of the cylinder, \( d = 1.60 \pm 0.01 \) cm
- Length (height) of the cylinder, \( h = 5.0 \pm 0.1 \) cm

### Step 2: Calculate the radius
The radius \( r \) is half of the diameter:
\[ r = \frac{d}{2} = \frac{1.60}{2} = 0.80 \, \text{cm} \]
The uncertainty in the radius \( \delta r \) is half the uncertainty in the diameter:
\[ \delta r = \frac{\delta d}{2} = \frac{0.01}{2} = 0.005 \, \text{cm} \]
Thus, we have:
\[ r = 0.80 \pm 0.005 \, \text{cm} \]

### Step 3: Write the formula for the volume of the cylinder
The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]

### Step 4: Calculate the percentage error in volume
The formula for the percentage error in volume is given by:
\[ \frac{\delta V}{V} \times 100 = 2 \frac{\delta r}{r} \times 100 + \frac{\delta h}{h} \times 100 \]
Where:
- \( \delta V \) is the uncertainty in volume
- \( \delta r \) is the uncertainty in radius
- \( \delta h \) is the uncertainty in height

### Step 5: Substitute the values into the formula
1. Calculate \( \frac{\delta r}{r} \):
\[ \frac{\delta r}{r} = \frac{0.005}{0.80} = 0.00625 \]

2. Calculate \( \frac{\delta h}{h} \):
\[ \frac{\delta h}{h} = \frac{0.1}{5.0} = 0.02 \]

3. Substitute these values into the percentage error formula:
\[ \text{Percentage error} = 2 \times 0.00625 \times 100 + 0.02 \times 100 \]
\[ = 1.25 + 2 = 3.25\% \]

### Step 6: Finalize the answer
The percentage error in the volume of the cylinder is:
\[ \text{Percentage error} = 3.25\% \]