The semi-vertical angle of a right circular cone is $45^{\circ}$. If the radius of the base of the cone is measured as 14 cm with an error of $\left(\frac{\sqrt{2}-1}{11}\right) \mathrm{cm}$, then the approximate error in measuring its total surface area is (in sq. cm )

• A

  $14$
• Correct answer
  B

  $8$
• C

  $5$
• D

  $3$

Solution

$\begin{aligned} & \quad \alpha=45^{\circ}, r=14 \mathrm{~cm}, l=\frac{r}{\sin 45^{\circ}}=14 \sqrt{2} \\ & \text { Total surface area }(\mathrm{A})=\pi \mathrm{r}(\mathrm{r}+l) \\ & \mathrm{A}=\pi r^2(1+\sqrt{2}) \\ & \frac{d \mathrm{~A}}{d r}=2 \pi r(1+\sqrt{2}) \\ & d A=2 \pi r(1+\sqrt{2}) d r \\ & d A=2 \pi(14)(1+\sqrt{2})\left(\frac{\sqrt{2}-1}{11}\right)=8\end{aligned}$

Hence, the correct answer is option: 'B'.