The length l , breadth b, and thickness `t` of a block of wood were measured with the help os a measuring scale. The results with permissible errors ( in cm) are
`l = 15.12 +-0.01, b = 10.15+-0.01` , and `t = 5.28 +- 0.01`.
The percentage error in volume up to proper significant figures is

Options

A. `0.28 %`
B. `0.35%`
C. `0.48%`
D. `0.64%`

Explanation

To find the percentage error in the volume of the block of wood, we can follow these steps:

### Step 1: Understand the formula for volume
The volume \( V \) of a block of wood is given by the formula:
\[ V = l \times b \times t \]
where \( l \) is the length, \( b \) is the breadth, and \( t \) is the thickness.

### Step 2: Calculate the permissible errors
The permissible errors in the measurements are given as:
- \( \Delta l = 0.01 \, \text{cm} \)
- \( \Delta b = 0.01 \, \text{cm} \)
- \( \Delta t = 0.01 \, \text{cm} \)

### Step 3: Calculate the volume and its absolute error
Using the measured values:
- \( l = 15.12 \, \text{cm} \)
- \( b = 10.15 \, \text{cm} \)
- \( t = 5.28 \, \text{cm} \)

First, calculate the volume \( V \):
\[ V = 15.12 \times 10.15 \times 5.28 \]

Now, calculate the absolute error in volume using the formula for the propagation of errors:
\[ \frac{\Delta V}{V} = \frac{\Delta l}{l} + \frac{\Delta b}{b} + \frac{\Delta t}{t} \]

### Step 4: Substitute the values into the error formula
Substituting the values:
\[ \frac{\Delta V}{V} = \frac{0.01}{15.12} + \frac{0.01}{10.15} + \frac{0.01}{5.28} \]

### Step 5: Calculate each term
Calculate each term:
1. \( \frac{0.01}{15.12} \approx 0.000661 \)
2. \( \frac{0.01}{10.15} \approx 0.000985 \)
3. \( \frac{0.01}{5.28} \approx 0.001893 \)

### Step 6: Sum the contributions
Now, sum these contributions:
\[ \frac{\Delta V}{V} \approx 0.000661 + 0.000985 + 0.001893 \approx 0.003539 \]

### Step 7: Convert to percentage
To find the percentage error, multiply by 100:
\[ \text{Percentage Error} = 0.003539 \times 100 \approx 0.3539\% \]

### Step 8: Round to proper significant figures
Rounding to three significant figures, we get:
\[ \text{Percentage Error} \approx 0.354\% \]

### Final Answer
The percentage error in volume is approximately \( 0.35\% \).

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