The radius of a sphere is measured as (2.1± 0.5) cm. Calculate its surface area with error limits.

The correct option is (55.4± 26.4) cm².

Step-by-step Explanation:

1. Find the surface area (A):
The formula for the surface area of a sphere is:
$A=4\pi r^2$
Given the measured radius is $r=2.1\text{cm}$ with an error limit $\Delta r=0.5\text{cm}$.
Using $\pi \approx \frac{22}{7}$:
$A=4\times \frac{22}{7}\times {\left(2.1\right)}^2$
$A=4\times \frac{22}{7}\times 4.41$
$A=4\times 22\times 0.63$
$A=88\times 0.63=55.44{\text{cm}}^2\approx 55.4{\text{cm}}^2$

2. Find the absolute error in surface area ($\Delta A$):
Taking the relative error for the formula $A=4\pi r^2$:
$\frac{\Delta A}{A}$
Rearranging to solve for $\Delta A$:
$\Delta A=2\times \frac{\Delta r}{r}\times A$
Substitute the values into the equation:
$\Delta A=2\times \frac{0.5}{2.1}\times 55.44$
$\Delta A=\frac{1}{2.1}\times 55.44$
$\Delta A=26.4{\text{cm}}^2$

3. Write final value with error limits:
Combining the area and the absolute error, we get:
$\text{Area}=\left(A\pm \Delta A\right)=\left(55.4\pm 26.4\right){\text{cm}}^2$