the angle of a sector of a circle with radius 22cm is 66 degrees. If the sector folded such that the straight edge coincides, forming a cone, calculate, correct to one decimal place, the radius, height and volume of the cube
To find the radius, height, and volume of the cone formed by folding the sector of a circle, we can follow these steps:
Calculate the arc length of the sector: The arc length ( L ) of a sector can be calculated using the formula: [ L = \frac{\theta}{360} \times 2\pi r ] where ( \theta ) is the angle in degrees and ( r ) is the radius of the circle.
Given:
Plugging in the values: [ L = \frac{66}{360} \times 2\pi \times 22 ] [ L = \frac{66}{360} \times 44\pi ] [ L = \frac{66 \times 44\pi}{360} ] [ L = \frac{2904\pi}{360} \approx 25.5 \text{ cm} \quad (\text{using } \pi \approx 3.14) ]
Determine the radius of the base of the cone: The arc length ( L ) becomes the circumference of the base of the cone when the sector is folded. The circumference ( C ) of a circle is given by: [ C = 2\pi r_{\text{base}} ] Setting ( C = L ): [ 2\pi r_{\text{base}} = L ] [ r_{\text{base}} = \frac{L}{2\pi} = \frac{25.5}{2\pi} \approx \frac{25.5}{6.28} \approx 4.06 \text{ cm} ]
Determine the slant height of the cone: The slant height ( l ) of the cone is equal to the radius of the original circle: [ l = 22 \text{ cm} ]
Calculate the height of the cone: We can use the Pythagorean theorem to find the height ( h ) of the cone: [ l^2 = r_{\text{base}}^2 + h^2 ] Rearranging gives: [ h^2 = l^2 - r_{\text{base}}^2 ] Plugging in the values: [ h^2 = 22^2 - 4.06^2 ] [ h^2 = 484 - 16.48 \approx 467.52 ] [ h \approx \sqrt{467.52} \approx 21.6 \text{ cm} ]
Calculate the volume of the cone: The volume ( V ) of a cone is given by: [ V = \frac{1}{3} \pi r_{\text{base}}^2 h ] Plugging in the values: [ V = \frac{1}{3} \pi (4.06)^2 (21.6) ] [ V \approx \frac{1}{3} \times 3.14 \times 16.48 \times 21.6 ] [ V \approx \frac{1}{3} \times 3.14 \times 355.68 \approx \frac{1114.56}{3} \approx 371.52 \text{ cm}^3 ]