Question

A chord of a circle of radius 14 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle.

OR

A solid is in the shape of a cone standing on a right circular cylinder with both their radii being equal to 7.5 cm and the height of the cone is equal to its radius. If the total height of the solid is 22.5 cm, find the volume of the solid. (Use $\pi = 3.14$, $\sqrt{3} = 1.732$)

Hint:

When $\theta = 60^\circ$, the triangle formed by the radii and the chord is always equilateral. If $\theta = 90^\circ$, use Area $= \frac{1}{2}r^2$.

Answer 1

Step 1: Understanding the Concept:
The area of a minor segment is calculated by subtracting the area of the triangle formed by the chord and radii from the area of the corresponding sector. The major segment area is then found by subtracting the minor segment area from the total area of the circle.

Step 2: Calculating Area of Sector and Triangle:
Given: \( r = 14 \) cm, \( \theta = 60^\circ \).
Area of Sector = \( \frac{\theta}{360} \times \pi r^2 \)
= \( \frac{60}{360} \times \frac{22}{7} \times 14 \times 14 \)
= \( \frac{1}{6} \times 22 \times 2 \times 14 \)
= \( 102.67 \text{ cm}^2 \)

Area of \( \triangle OAB \): Since \( \theta = 60^\circ \) and \( OA = OB \), the triangle is equilateral.
Area = \( \frac{\sqrt{3}}{4} r^2 \)
= \( \frac{1.732}{4} \times 14 \times 14 \)
= \( 1.732 \times 49 \)
= \( 84.868 \text{ cm}^2 \)

Step 3: Calculating Minor Segment Area:
Area of Minor Segment = Area of Sector − Area of Triangle
= \( 102.67 - 84.868 \)
= \( 17.802 \text{ cm}^2 \)

Step 4: Calculating Major Segment Area and Final Answer:
Area of Circle = \( \pi r^2 \)
= \( \frac{22}{7} \times 14 \times 14 \)
= \( 616 \text{ cm}^2 \)
Area of Major Segment = \( 616 - 17.802 \)
= \( 598.198 \text{ cm}^2 \)
The area of the minor segment is \( 17.80 \) cm² and the major segment is \( 598.20 \) cm².

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Solution (OR Question):

Step 1: Understanding the Concept:
The volume of the solid is the sum of the volume of the cone and the volume of the cylinder. We must first determine the individual heights of both parts.

Step 2: Identifying Dimensions:
Radius \( r \) for both = 7.5 cm
Height of cone \( h_1 \) = 7.5 cm
Total height of solid = 22.5 cm
Height of cylinder \( h_2 = 22.5 - 7.5 = 15 \) cm

Step 3: Volume Calculation:
Volume of solid = Volume of Cone + Volume of Cylinder
= \( \frac{1}{3}\pi r^2 h_1 + \pi r^2 h_2 \)
= \( \pi r^2 \left(\frac{1}{3}h_1 + h_2\right) \)
= \( 3.14 \times (7.5)^2 \times \left(\frac{7.5}{3} + 15\right) \)
= \( 3.14 \times 56.25 \times (2.5 + 15) \)
= \( 176.625 \times 17.5 \)

Step 4: Final Answer:
Volume = \( 3090.9375 \text{ cm}^3 \)
The volume of the solid is \( 3090.94 \text{ cm}^3 \) (approx).