Question:
Rekha drew a circle of radius 2 cm on a graph paper of grid 1 cm × 1 cm. She then calculated the area of the circle by adding up only the number of full unit-squares that fell within the perimeter of the circle. If the value that Rekha obtained was 4 sq cm less than the correct value, then find the minimum possible value of \(d\).
Rekha drew a circle of radius 2 cm on a graph paper of grid 1 cm × 1 cm. She then calculated the area of the circle by adding up only the number of full unit-squares that fell within the perimeter of the circle. If the value that Rekha obtained was 4 sq cm less than the correct value, then find the minimum possible value of \(d\).
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Area underestimation from digitization often yields approximations closer to integer-covered regions. Estimate the reduced area directly.
Updated On: Jul 28, 2025
- 6.28
- 7.28
- 7.56
- 8.56
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The Correct Option is D
Solution and Explanation
Step 1: Correct area of circle
Radius \(r = 2\) cm. \[ \text{Area} = \pi r^2 = \pi \cdot 4 \approx 12.56 \text{ sq cm} \] Step 2: Rekha’s estimated area
She underestimated by 4 sq cm, so: \[ \text{Her estimate} = 12.56 - 4 = 8.56 \text{ sq cm} \] Since the circle is being approximated using unit squares, and this estimation is dependent on resolution of grid (i.e., grid size = 1 cm), the effective resolution error is related to the perimeter. Step 3: Area underestimation indicates coarser circle sampling.
We must estimate what value of \(d\) could result in an estimate of 8.56. Since radius is fixed, this is a distractor: The correct value is the under-estimate Rekha got: \[ \boxed{8.56} \]
Radius \(r = 2\) cm. \[ \text{Area} = \pi r^2 = \pi \cdot 4 \approx 12.56 \text{ sq cm} \] Step 2: Rekha’s estimated area
She underestimated by 4 sq cm, so: \[ \text{Her estimate} = 12.56 - 4 = 8.56 \text{ sq cm} \] Since the circle is being approximated using unit squares, and this estimation is dependent on resolution of grid (i.e., grid size = 1 cm), the effective resolution error is related to the perimeter. Step 3: Area underestimation indicates coarser circle sampling.
We must estimate what value of \(d\) could result in an estimate of 8.56. Since radius is fixed, this is a distractor: The correct value is the under-estimate Rekha got: \[ \boxed{8.56} \]
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