Question

If the length and breadth of a plane are $(40 + 0.2)$ and $(30 \pm 0.1)$ cm, the absolute error in the measurement of area is:

A. $10 ~\text{cm}^2$

B. $8 ~\text{cm}^2$

C. $9 ~\text{cm}^2$

D. $7 ~\text{cm}^2$

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Answer

To solve for the absolute error in the area given the length and breadth with their respective errors, we can follow the steps below:

Given:

  • Length ($L \pm \Delta L$) = $40 \pm 0.2$ cm

  • Breadth ($B \pm \Delta B$) = $30 \pm 0.1$ cm

Steps to :

  1. Calculate the area without errors: The nominal area ($A$) is calculated by: $$ A = L \times B = 40 \times 30 = 1200 \ \text{cm}^2 $$

  2. Understand the error in areas: The absolute error in area ($\Delta A$) can be determined using the formula: $$ \frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} $$

  3. Substitute the given values:

    • $\Delta L = 0.2$ cm

    • $L = 40$ cm

    • $\Delta B = 0.1$ cm

    • $B = 30$ cm

    Using formula: $$ \frac{\Delta A}{1200} = \frac{0.2}{40} + \frac{0.1}{30} $$

  4. Calculate the terms on the right-hand side:

    • $\frac{0.2}{40} = 0.005$

    • $\frac{0.1}{30} \approx 0.00333$

    Adding these: $$ \frac{\Delta A}{1200} = 0.005 + 0.00333 = 0.00833 $$

  5. Calculate the absolute error: $$ \Delta A = 1200 \times 0.00833 \approx 10 \ \text{cm}^2 $$

Final Answer

The absolute error in the measurement of the area is 10 cm².

So, the correct option is Option A.


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