12. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle to a distance of . Find the area of the sea over which the ships are warned. (Use )

Understand the Question The light from the lighthouse sweeps out a sector of a circle. Given: Central angle of the sector Radius Value of The area of the sea warned corresponds to the area of this sector, given by the formula: Step 1 · Calculate the Area of the Sector Given radius and sector angle . Substituting , , and Answer > Common Mistakes Incorrect Value of : Using instead of the explicitly specified , which leads to slight decimal inaccuracies. Unit Errors: Forgetting to write the square unit for area, or writing instead. Formula Confusion: Confusing the area of a sector with arc length .

Areas Related to Circles | Exercise 11.1

Question 12

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 $\pi = 3.14$ )

Solution

Understand the Question

The light from the lighthouse sweeps out a sector of a circle.
Given:
- Central angle of the sector $(\theta) = 80^\circ$
- Radius $(r) = 16.5\text{ km}$
- Value of $\pi = 3.14$
The area of the sea warned corresponds to the area of this sector, given by the formula: $\text{Area of Sector} = \dfrac{\theta}{360^\circ} \times \pi r^2$

Step 1 · Calculate the Area of the Sector

Given radius $r = 16.5\text{ km}$ and sector angle $\theta = 80^\circ$ . Diagram 1

$\text{Area of sector} = \dfrac{\theta}{360^\circ} \times \pi r^2$

Substituting $\theta = 80^\circ$ , $r = 16.5\text{ km}$ , and $\pi = 3.14$

\begin{aligned} \text{Area} &= \dfrac{80^\circ}{360^\circ} \times 3.14 \times (16.5)^2 \\[0.6em] &= \dfrac{2}{9} \times 3.14 \times 16.5 \times 16.5 \\[0.6em] &= \dfrac{2}{9} \times 3.14 \times 272.25 \\[0.6em] &= \dfrac{1709.73}{9} \\[0.6em] &= 189.97\text{ km}^2 \end{aligned}

Answer

$189.97\text{ km}^2$

Common Mistakes

Incorrect Value of $\pi$ : Using $\pi = \dfrac{22}{7}$ instead of the explicitly specified $\pi = 3.14$ , which leads to slight decimal inaccuracies.
Unit Errors: Forgetting to write the square unit $(\text{km}^2)$ for area, or writing $\text{km}$ instead.
Formula Confusion: Confusing the area of a sector $\left(\dfrac{\theta}{360^\circ} \times \pi r^2\right)$ with arc length $\left(\dfrac{\theta}{360^\circ} \times 2\pi r\right)$ .

More questions in Exercise 11.1

Unless stated otherwise, use $\pi = \dfrac{22}{7}$ .

Find the area of a sector of a circle with radius $6 \text{ cm}$ if angle of the sector is $60^\circ$ .

Find the area of a quadrant of a circle whose circumference is $22\text{ cm}$ .

The length of the minute hand of a clock is $14\text{ cm}$ . Find the area swept by the minute hand in $5$ minutes.

A chord of a circle of radius $10\text{ cm}$ subtends a right angle at the centre. Find the area of the corresponding :

(i) minor segment

(ii) major sector. (Use $\pi = 3.14$ )

In a circle of radius $21 \text{ cm}$ , an arc subtends an angle of $60^\circ$ at the centre. Find:

(i) the length of the arc

(ii) area of the sector formed by the arc

(iii) area of the segment formed by the corresponding chord

A chord of a circle of radius $15\text{ cm}$ subtends an angle of $60^\circ$ at the centre. Find the areas of the corresponding minor and major segments of the circle.

(Use $\pi = 3.14$ and $\sqrt{3} = 1.73$ )

A chord of a circle of radius $12\text{ cm}$ subtends an angle of $120^\circ$ at the centre. Find the area of the corresponding segment of the circle.

(Use $\pi = 3.14$ and $\sqrt{3} = 1.73$ )

A horse is tied to a peg at one corner of a square shaped grass field of side $15\text{ m}$ by means of a $5\text{ m}$ long rope (see Fig. 11.8). Find

(i) the area of that part of the field in which the horse can graze.

(ii) the increase in the grazing area if the rope were $10\text{ m}$ long instead of $5\text{ m}$ . (Use $\pi = 3.14$ )

A brooch is made with silver wire in the form of a circle with diameter $35\text{ mm}$ . The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in Fig. 11.9. Find :

(i) the total length of the silver wire required.

(ii) the area of each sector of the brooch.

Q10

An umbrella has 8 ribs which are equally spaced (see Fig. 11.10). Assuming umbrella to be a flat circle of radius $45\text{ cm}$ , find the area between the two consecutive ribs of the umbrella.

Q11

A car has two wipers which do not overlap. Each wiper has a blade of length $25\text{ cm}$ sweeping through an angle of $115^\circ$ . Find the total area cleaned at each sweep of the blades.

Q12

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 $\pi = 3.14$ )

Q13

A round table cover has six equal designs as shown in Fig. 11.11. If the radius of the cover is $28\text{ cm}$ , find the cost of making the designs at the rate of ₹ $0.35$ per $\text{cm}^2$ . (Use $\sqrt{3} = 1.7$ )

Q14

Tick the correct answer in the following :

Area of a sector of angle $p$ (in degrees) of a circle with radius $R$ is

(A) $\dfrac{p}{180} \times 2\pi R$ (B) $\dfrac{p}{180} \times \pi R^2$ (C) $\dfrac{p}{360} \times 2\pi R$ (D) $\dfrac{p}{720} \times 2\pi R^2$