Question

To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle \[{80^\circ }\] to a distance of $16.5{\text{ km}}$. Find the area of the sea over which the ships are warned. (Use π=3.14)

Answer 1

Hint: In this question we will use the concept of the sector of a circle and its area. Here, we have given some parameters in the question like radius and angle subtended so, we will use these details to find out the area of the sea over which the ships are warned and that area is similar to the area of a sector with angle $\theta $ and we get the required answer.

Complete step-by-step solution -

If the arc subtends an angle$\theta $, then the area of the corresponding sector is, $\dfrac{\theta }{{360}} \times \pi {r^2}$.
Given that , radius = $16.5{\text{ km}}$ and subtended angle = \[{80^\circ }\] and use $\pi = 3.14$.
Now we have to find area over which ships are warned ,
The arc over which ships are warned is equal to the area of the sector with angle \[{80^\circ }\].
$
   \Rightarrow {\text{ area of sector with angle 80}}^\circ {\text{ = }}\dfrac{\theta }{{360}} \times \pi {r^2} \\
  {\text{ = }}\dfrac{{{\text{80}}^\circ }}{{360}} \times \pi \times {\left( {16.5} \right)^2} \\
  {\text{ = }}\dfrac{{{\text{80}}^\circ }}{{360}} \times 3.14 \times 16.5 \times 16.5 \\
  {\text{ = }}\dfrac{2}{9} \times \dfrac{{314}}{{100}} \times \dfrac{{165}}{{10}} \times \dfrac{{165}}{{10}} \\
  {\text{ = 189}}{\text{.97k}}{{\text{m}}^2}. \\
 $
Hence, the area of the sea over which the ships are warned = ${\text{189}}{\text{.97k}}{{\text{m}}^2}.$

Note: In this type of question first we have to know what is given in the question and what we have to find . here we have to find the area of the sector , so we used the formula to find the area of the sector when the radius of sector and the angle subtended is given . hence, by putting these values simply in the formula we will get the required answer.