Question

The radius of a sphere is measured to be $\left( 2.1\pm 0.5 \right)$cm. Calculate its surface area with error limits.

Answer 1

The formula to find error limits is to be used in questions like these. Here, we are given a sphere and the formula to find the surface area of the sphere is known to us. Hence, the students are supposed to combine both these results to find the required value within the error limits.

Formulas used:
The surface area of sphere $A=4\pi {{r}^{2}}$
The formula for relative error to find the absolute error is found by $\dfrac{\Delta A}{A}=2\times \dfrac{\Delta r}{r}$

Complete step by step solution:
We use error limits while performing calculations because during experimentation or calculation, certain errors can be caused due to flaws in the measuring instrument or due to calculation mistakes. Error limits help in preventing these issues. It is nothing but the maximum and minimum deviation from the final value that is to be gained.We know that surface area of sphere $A=4\pi {{r}^{2}}$.
Hence Area=$4\times 3.14\times {{(2.1)}^{2}}=55.4c{{m}^{2}}$.
The formula for relative error to find the absolute error can be calculated by
$\dfrac{\Delta A}{A}=2\times \dfrac{\Delta r}{r}$
Which is,
$\begin{aligned} & \Delta A=2\times \dfrac{\Delta r}{r}\times A \\\ &\Rightarrow\Delta A =2\times \dfrac{0.5}{2.1}\times 55.4 \\\ &\therefore\Delta A =26.4c{{m}^{2}} \\\ \end{aligned}$
Now to find the surface area within the error limits, we must express the given values in the form $(A\pm \Delta A)$.

Thus the surface area of the given sphere within error limits is $(55.4\pm 26.4)c{{m}^{2}}$.

Note: The concept of relative errors and absolute errors is to be noted. Relative error is the ratio between the absolute error and the mean value of the quantity that is provided. Whereas absolute error is nothing but the difference between the measured value and the true value of the quantity.