Ex.12.2 Q12 Areas Related to Circles Solution - NCERT Maths Class 10

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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 \(\text{16.5 km.}\) Find the area of the sea over which the ships are warned.

(Use \(\begin{align}\pi=3.14\end{align}\))

Text Solution

What is known?

A lighthouse spreads a red coloured light over a sector of angle of \(\begin{align}80^{\circ}\end{align}\) to a distance of \(16.5 \,\rm{km}\) to warn ships for underwater rock.

(Use \(\begin{align}\pi=3.14\end{align}\))

What is unknown?

Area of the sea over which the ships are warned.

Reasoning:

Since the lighthouse spreads red coloured light over a sector  of a circle with radius \(\text{= 16.5 km}\) and angle with degree measure \(\begin{align}80^{\circ}\end{align}\).

Area of sea over which the ship are warned = area of the sector of the circle with radius 16.5km and angle with degree measure\(\begin{align}80^{\circ}\end{align}\).

\[\begin{align}&={\frac{\theta}{360^{\circ}} \times \pi r^{2}} \\ &={\frac{80^{\circ}}{360^{\circ}} \times \pi r^{2}} \\ &={\frac{2}{9} \times \pi r^{2}} \end{align}\]

Steps:

Area of sea over which the ships are warned = Area of a sector of the circle with radius, \(\rm{}r=16.5\,km\) and angle with degree measure\(\begin{align}80^{\circ}\end{align}\).

\[\begin{align} &= \frac{{{{80}^\circ }}}{{{{360}^\circ }}} \times \pi {r^2}\\ &= \frac{2}{9} \times 3.14 \times 16.5 \times 16.5\\ &= 189.97\,\,{\text{k}}{{\text{m}}^2}\end{align}\]