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

Solution 1

Step 1: Write the given information.
Radius of the circle \(r = 14\) cm.
Angle subtended by the chord at the centre \(= 60^\circ\).

Step 2: Find the area of the sector.
Area of a sector \(=\frac{\theta}{360^\circ}\pi r^2\).
\[ \text{Area of sector}=\frac{60}{360}\times \pi \times 14^2 \] \[ =\frac{1}{6}\times \pi \times 196 \] Using \(\pi=\frac{22}{7}\): \[ =\frac{1}{6}\times \frac{22}{7}\times 196 \] \[ =\frac{616}{3}\approx 205.33 \text{ cm}^2 \]

Step 3: Find the area of triangle formed by the radii.
The triangle formed by the two radii and the chord has sides \(14,14\) and included angle \(60^\circ\).
Area of triangle \(=\frac{1}{2}ab\sin C\). \[ =\frac{1}{2}\times 14 \times 14 \times \sin60^\circ \] \[ =98\times \frac{\sqrt3}{2} \] \[ =49\sqrt3 \] Using \(\sqrt3=1.732\): \[ =49\times 1.732 \] \[ \approx 84.87 \text{ cm}^2 \]

Step 4: Find the area of the minor segment.
\[ \text{Minor segment} = \text{sector} - \text{triangle} \] \[ =205.33 - 84.87 \] \[ \approx 120.46 \text{ cm}^2 \]

Step 5: Find the area of the major segment.
Area of the whole circle: \[ \pi r^2=\frac{22}{7}\times 196=616 \] \[ \text{Major segment} = 616 - 120.46 \] \[ \approx 495.54 \text{ cm}^2 \]

Final Answer:
Area of the minor segment \( \approx 120.46 \text{ cm}^2 \).
Area of the major segment \( \approx 495.54 \text{ cm}^2 \).

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Solution 2

Step 1: Write the given data.
Radius of cylinder and cone \(r = 7.5\) cm.
Height of cone \(= r = 7.5\) cm.
Total height of the solid \(= 22.5\) cm.

Step 2: Find the height of the cylinder.
\[ \text{Height of cylinder} = 22.5 - 7.5 \] \[ = 15 \text{ cm} \]

Step 3: Find the volume of the cylinder.
\[ V = \pi r^2 h \] \[ = 3.14 \times (7.5)^2 \times 15 \] \[ = 3.14 \times 56.25 \times 15 \] \[ = 3.14 \times 843.75 \] \[ \approx 2649.38 \text{ cm}^3 \]

Step 4: Find the volume of the cone.
\[ V = \frac{1}{3}\pi r^2 h \] \[ = \frac{1}{3}\times 3.14 \times (7.5)^2 \times 7.5 \] \[ = \frac{1}{3}\times 3.14 \times 421.875 \] \[ \approx 441.56 \text{ cm}^3 \]

Step 5: Find the total volume of the solid.
\[ \text{Total volume} = 2649.38 + 441.56 \] \[ \approx 3090.94 \text{ cm}^3 \]

Final Answer:
The volume of the solid \( \approx 3090.94 \text{ cm}^3 \).