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The length of a rectangular plate is mea...

The length of a rectangular plate is measured as 10 cm by a vernier scale of least count 0.01 cm and its breadth as 5 cm by the same scale. The percentage error in area is

A

`0.1%`

B

`0.3 %`

C

`0.01%`

D

`0.05%`

Text Solution

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The correct Answer is:
To find the percentage error in the area of a rectangular plate, we can follow these steps: ### Step 1: Understand the formula for area The area \( A \) of a rectangle is given by the formula: \[ A = \text{Length} \times \text{Breadth} \] ### Step 2: Identify the measurements From the problem, we have: - Length \( L = 10 \) cm - Breadth \( B = 5 \) cm ### Step 3: Determine the least count The least count of the vernier scale is given as: \[ \text{Least Count} = 0.01 \text{ cm} \] ### Step 4: Calculate the absolute errors in measurements The absolute error in length \( \Delta L \) and breadth \( \Delta B \) can be taken as the least count: \[ \Delta L = 0.01 \text{ cm} \] \[ \Delta B = 0.01 \text{ cm} \] ### Step 5: Calculate the relative errors The relative error in length \( \frac{\Delta L}{L} \) and breadth \( \frac{\Delta B}{B} \) can be calculated as follows: \[ \frac{\Delta L}{L} = \frac{0.01}{10} = 0.001 \] \[ \frac{\Delta B}{B} = \frac{0.01}{5} = 0.002 \] ### Step 6: Calculate the total relative error in area The total relative error in area \( \frac{\Delta A}{A} \) is given by the sum of the relative errors in length and breadth: \[ \frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} \] Substituting the values: \[ \frac{\Delta A}{A} = 0.001 + 0.002 = 0.003 \] ### Step 7: Convert relative error to percentage error To find the percentage error, we multiply the relative error by 100: \[ \text{Percentage Error} = \frac{\Delta A}{A} \times 100 = 0.003 \times 100 = 0.3\% \] ### Final Result The percentage error in the area of the rectangular plate is: \[ \text{Percentage Error} = 0.3\% \]
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