A sector of a circle radius 8cm subtends and and angle 90°a at the centre of a circle. If the sector is folded without overlap to form the curved surface of a cone, find the 1. Base radius 2. Height 3. Volume of the cone.
It is not well explanatory
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it's not well explanatory
To find the base radius, height, and volume of the cone formed by folding the sector, we need to apply some formulas and trigonometric principles. Let's tackle each part separately:
1. To find the base radius:
- The sector subtends an angle of 90° at the center of the circle, which means it covers one-quarter (1/4) of the entire circumference (360°) of the circle.
- The circumference of a circle is given by the formula: circumference = 2 * π * radius.
- Since the sector covers 1/4 of the circumference, the arc length of the sector can be calculated using: arc length = (1/4) * circumference.
- In this case, the arc length is equal to the curved portion of the cone's base perimeter.
- So, we can set up the equation: arc length = 2π * base radius of the cone.
- Substituting the given values, we have: (1/4) * 2π * 8 = 2π * base radius.
- Simplifying the equation, we get: π * 4 = 2π * base radius.
- Dividing both sides by 2π, we find: base radius = 2 cm.
Therefore, the base radius of the cone is 2 cm.
2. To find the height:
- When the sector is folded and joined, it forms the lateral surface of a cone.
- The length of the lateral surface of a cone is equal to the circumference of the base of the cone.
- We know the circumference of the base is 2π * base radius = 2π * 2 = 4π cm.
- The height of the cone is the slant height of this lateral surface, which acts as the slant height of the folded sector.
- To find the slant height, we can use the Pythagorean theorem, which states that the square of the hypotenuse (slant height) is equal to the sum of the squares of the other two sides.
- In this case, we have a right-angled triangle with one side as the radius of the circle (8 cm) and the other side as the base radius of the cone (2 cm).
- The slant height can be calculated as: hypotenuse = √(radius^2 + base radius^2).
- Substituting the given values, we get: hypotenuse = √(8^2 + 2^2) = √(64 + 4) = √68 ≈ 8.246 cm.
Therefore, the height of the cone is approximately 8.246 cm.
3. To find the volume of the cone:
- The volume of a cone can be calculated using the formula: volume = (1/3) * π * base radius^2 * height.
- Substituting the known values, we have: volume = (1/3) * π * 2^2 * 8.246.
- Simplifying the equation, we find: volume ≈ 54.76 cm^3.
Therefore, the volume of the cone is approximately 54.76 cm^3.
s = rθ so the circumference of the base of the cone is 8*π/2 = 4π
Thus the radius of the base is 4π/2π = 2 (1/4 the radius of the original circle)
The slant height is the original circle radius.
so the height of the cone is √(8^2-2^2) = √60
as always, the volume v = 1/3 πr^2 h