In the diagram AOBP is the major sector of a circle centre O.BO=8cm, AOB=156° and the sector is bent to form a right cone.
a)Caculate the radius of the cone
b)Calculate the vertical angle of the cone
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An angle of 216 sector of a circle of radius 5 cm is bent to form a cone. Find the radius of the base of the cone and its vertical angle?
To calculate the radius of the cone formed from the major sector, we can use the formula for the circumference of a circle:
C = 2πr
In the given diagram, we know that the arc length AOB is equal to the circumference of the base of the cone.
The arc length can be calculated using the formula for the circumference of a sector:
S = (θ/360) × 2πr
In this case, θ = 156° and S = 8 cm. Plugging the values into the formula:
8 = (156/360) × 2πr
To solve for r, we can rearrange the equation:
r = (8 × 360) / (156 × 2π)
r = 10.91 cm
Therefore, the radius of the cone is approximately 10.91 cm.
To calculate the vertical angle of the cone, we can use the trigonometric relationship between the slant height (l) and the radius (r) of the cone:
sin(θ) = r / l
Given that the radius (r) is 10.91 cm, we need to find the slant height (l) to calculate the vertical angle (θ).
We can use Pythagoras' theorem to find the slant height:
l^2 = r^2 + h^2
Since the cone is a right cone, the height (h) is equal to the radius (r). Plugging in the values:
l^2 = (10.91)^2 + (10.91)^2
l^2 = 237.84
l = √237.84
l ≈ 15.43 cm
Now, we can calculate the vertical angle (θ) using the formula:
θ = arcsin(r / l)
θ = arcsin(10.91 / 15.43)
θ ≈ 45.84°
Therefore, the vertical angle of the cone is approximately 45.84°.