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:
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First calculate \( 2 \times \frac{22}{7} \): \[ 2 \times \frac{22}{7} = \frac{44}{7} \]
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Now calculate \( 2\pi r = \frac{44}{7} \times 40 \): \[ = \frac{1760}{7} \text{ cm} \]
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Now compute \( \frac{126}{360} \): \[ \frac{126}{360} = \frac{7}{20} \]
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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:
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First calculate \( \pi r^2 \): \[ \pi r^2 = \frac{22}{7} \times (40^2) = \frac{22}{7} \times 1600 = \frac{35200}{7} \text{ cm}^2 \]
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Now compute the area of the sector: \[ A = \frac{126}{360} \times \frac{35200}{7} \]
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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:
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\( 14^2 = 196 \)
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So: \[ V = \frac{1}{3} \times \frac{22}{7} \times 196 \times 37.5 \]
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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)
- Length of the arc: 88 cm
- Area of the sector: 1760 cm²
(b)
- Radius of the base of the cone: 14 cm
- Height of the cone: 37.5 cm
- Capacity of the cone: 7.7 liters