Question

Arc of a circle of radius 40cm subtens an angle of 126 degrees at the center of the circle.(take pi=22/7)
(a) calculate
(I) the length of the arc
(Ii) the area of the sector
(b) the sector is folded to form a cone. Calculate
(I) the radius of the base of the cone
(Ii) the height of the cone
(III) the capacity of the cone in litres

Answer 1

To solve the problem, we will use the given radius of the circle (\( r = 40 \) cm) and the angle subtended at the center of the circle (\( \theta = 126^\circ \)). We'll also use \(\pi \approx \frac{22}{7}\).

### (a) Calculate

#### (I) The length of the arc
The formula for the length of an arc (\( L \)) is given by:

\[
L = \frac{\theta}{360^\circ} \times 2\pi r
\]

Substituting the values:

\[
L = \frac{126}{360} \times 2 \times \frac{22}{7} \times 40
\]

Calculating:

1. First calculate \( 2 \times \frac{22}{7} \):
\[
2 \times \frac{22}{7} = \frac{44}{7}
\]

2. Now calculate \( 2\pi r = \frac{44}{7} \times 40 \):
\[
= \frac{1760}{7} \text{ cm}
\]

3. Now compute \( \frac{126}{360} \):
\[
\frac{126}{360} = \frac{7}{20}
\]

4. Finally compute the length of the arc using:
\[
L = \frac{7}{20} \times \frac{1760}{7}
\]
\[
= \frac{1760}{20} = 88 \text{ cm}
\]

So, the length of the arc is **88 cm**.

#### (II) The area of the sector
The formula for the area of the sector (\( A \)) is given by:

\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]

Substituting the values:

1. First calculate \( \pi r^2 \):
\[
\pi r^2 = \frac{22}{7} \times (40^2) = \frac{22}{7} \times 1600 = \frac{35200}{7} \text{ cm}^2
\]

2. Now compute the area of the sector:
\[
A = \frac{126}{360} \times \frac{35200}{7}
\]

3. Using \( \frac{126}{360} = \frac{7}{20} \):
\[
A = \frac{7}{20} \times \frac{35200}{7}
\]
\[
= \frac{35200}{20} = 1760 \text{ cm}^2
\]

So, the area of the sector is **1760 cm²**.

### (b) The sector is folded to form a cone.

When the sector is folded to form a cone, the lengths of the arc become the circumference of the base of the cone.

#### (I) The radius of the base of the cone
The circumference (\( C \)) of the base of the cone can be computed using the length of the arc:

\[
C = L = 88 \text{ cm}
\]

The radius (\( r_{\text{cone}} \)) of the base of the cone is given by:

\[
C = 2 \pi r_{\text{cone}}
\]

Rearranging the formula gives:

\[
r_{\text{cone}} = \frac{C}{2 \pi} = \frac{88}{2 \times \frac{22}{7}} = \frac{88 \times 7}{44} = 14 \text{ cm}
\]

So, the radius of the base of the cone is **14 cm**.

#### (II) The height of the cone

The height can be found using the Pythagorean theorem, where the slant height (\( l \)) is the radius of the original circle, and the base radius is known:

\[
l = 40 \text{ cm}, \quad r_{\text{cone}} = 14 \text{ cm}
\]

Using the Pythagorean theorem:

\[
l^2 = r_{\text{cone}}^2 + h^2
\]

Substituting the values:

\[
40^2 = 14^2 + h^2
\]
\[
1600 = 196 + h^2
\]
\[
h^2 = 1600 - 196 = 1404
\]
\[
h = \sqrt{1404} \approx 37.5 \text{ cm}
\]

So, the height of the cone is approximately **37.5 cm**.

#### (III) The capacity of the cone in liters

The volume (\( V \)) of a cone is given by:

\[
V = \frac{1}{3} \pi r_{\text{cone}}^2 h
\]

Substituting the values:

\[
V = \frac{1}{3} \times \frac{22}{7} \times (14^2) \times 37.5
\]

Calculating:

1. \( 14^2 = 196 \)
2. So:
\[
V = \frac{1}{3} \times \frac{22}{7} \times 196 \times 37.5
\]

3. Calculate:
\[
V = \frac{1}{3} \times \frac{22 \times 196 \times 37.5}{7}
\]
\[
= \frac{22 \times 7350}{21} = \frac{161700}{21} = 7700 \text{ cm}^3
\]

To convert cm³ to liters:

\[
\text{Volume in liters} = \frac{7700}{1000} = 7.7 \text{ liters}
\]

So, the capacity of the cone is **7.7 liters**.

### Summary of Results:
(a)
1. Length of the arc: **88 cm**
2. Area of the sector: **1760 cm²**

(b)
1. Radius of the base of the cone: **14 cm**
2. Height of the cone: **37.5 cm**
3. Capacity of the cone: **7.7 liters**