Question

The radius of a sphere is measured as (2.1 +- 0.5) cm calculate its surface area with error limits.
I got the answer but then why are we multiplying 2 to #r/r please solve my doubt ​

Answer 1

$\huge{\underline{\tt{\red{Answer-}}}}$

$Radius = (2.1 \pm 0.5) cm$

$Surface \ area = 4 \pi r^2$

                   =  $4\times 3.14 \times (2.1)^2$

                   = $55.4 cm^2$

Now,

$\dfrac{\Delta A}{A}= 2 \dfrac{\Delta r }{r}$

$\Delta A = A \times 2 \times \dfrac {\Delta r}{2.1}}$

     = $5.54 \times 2 \times \dfrac{0.5}{2.1}$

     =$26.4 cm ^2$

$\therefore$ surface area = $(55.4 \pm 26.4) \ cm^2$

Answer 2

Solution:

Given:

$\sf{a\pm\Delta a=(2.1\pm0.5) \ cm}$

$\sf{Here \ a=Radius \ of \ sphere,}$

$\sf{\Delta a=Error}$

To find:

1. Total surface area of sphere.
2. Error limits.

Formula:

$\sf{Limit \ error=Relative \ error}$

$\sf{(1) \ Relative \ error=\dfrac{\Delta a}{a}}$

Calculation:

$\boxed{\sf{Total \ surface \ area \ of \ sphere=4\pi\times \ r^{2}}}$

$\sf{\therefore{Total \ surface \ area \ of \ sphere=4\times\dfrac{22}{7}\times2.1^{2}}}$

$\sf{\therefore{Total \ surface \ area \ of \ sphere=55.44 \ cm^{2}}}$

$\sf{From \ formula \ (1)}$

$\sf{Limit \ error=\dfrac{\Delta a}{a}}$

$\sf{=2 \ \dfrac{0.5}{2.1}}$

$\sf{=2 \ \dfrac{1\times5}{2.1\times10}}$

$\sf{=0.4762}$

$\sf{=0.476 (approx)}$

$\sf\purple{\tt{\therefore{Total \ surface \ area \ of \ sphere}}}$

$\sf\purple{\tt{is \ 55.44 \ cm^{2} \ and \ it's \ limit \ error}}$

$\sf\purple{\tt{is \ 0.467 \ cm}}$