A sector of a circle of radius 8cm is bent to form a cone.find d radius of the cone and it's vertical angle if d angle subtended at d center by the sector is 280
The cone has base radius 280/360 * 8 = 5.89
The cone has height 8
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To find the radius of the cone, we can use the formula that relates the circumference of a circle to its radius:
C = 2πr
Since the sector is bent to form a cone, the length of the curved edge of the sector is equal to the circumference of the base of the cone.
Length of curved edge of sector = Circumference of cone base
The length of the curved edge of the sector is given by the formula:
Length of curved edge of sector = rθ
Where r is the radius of the circle, and θ is the angle subtended at the center by the sector. In this case, r = 8cm and θ = 280°.
Length of curved edge of sector = 8cm * (280°/360°) [Converting degree to radians]
Length of curved edge of sector = 8cm * (4π/9)
Next, we equate the length of the curved edge of the sector to the circumference of the cone base:
Length of curved edge of sector = Circumference of cone base
8cm * (4π/9) = 2π * cone base radius
Canceling out π from both sides:
8cm * (4/9) = 2 * cone base radius
Simplifying:
16/9 = cone base radius
Therefore, the radius of the cone is 16/9 cm.
To find the vertical angle of the cone, we use the formula that relates the angle of the sector to the slant height and radius of the cone:
θ = 2 * arcsin (r / slant height)
In this case, r = 16/9 cm and θ = 280°.
θ = 2 * arcsin (16/9 / slant height)
θ = 2 * arcsin (16/9 / r + height)
We know that the slant height of the cone is equal to the radius of the sector, which is 8cm.
θ = 2 * arcsin (16/9 / 8)
Simplifying:
θ = 2 * arcsin (2/9)
Using a calculator to find the inverse sine of 2/9:
θ ≈ 3.86°
Therefore, the vertical angle of the cone is approximately 3.86°.
To find the radius of the cone and its vertical angle, let's break down the problem into steps:
Step 1: Find the circumference of the circle sector.
The circumference of a circle is given by the formula C = 2πr, where r is the radius. In this case, the radius of the circle is given as 8cm. Hence, C = 2π(8) = 16π cm.
Step 2: Find the length of the arc of the sector.
The length of the arc of a sector is given by the formula L = (θ/360) × C, where θ is the angle in degrees and C is the circumference of the circle. The angle θ is given as 280 degrees. Substituting the given values, L = (280/360) × 16π = 4π cm.
Step 3: Set up an equation for the curved surface area of the cone.
The curved surface area (CSA) of a cone is given by the formula CSA = πrℓ, where r is the radius of the base of the cone and ℓ is the slant height. In this case, we know that the length of the arc is equal to the circumference of the base of the cone. Therefore, ℓ = L = 4π cm. We also want to find the radius (r) of the cone.
Step 4: Find the radius (r) of the cone.
Substituting the values into the equation CSA = πrℓ, we get CSA = πr(4π) = 4π^2r. The curved surface area of the cone is also equal to the area of the circle sector, which is equal to (θ/360) × πr^2. Substituting the given angle θ = 280 degrees, we get (280/360) × πr^2. Therefore, we can set up the equation (280/360) × πr^2 = 4π^2r.
Simplifying the equation, we have (7/9) × r^2 = 4πr.
Step 5: Solve the equation for the radius (r) of the cone.
To solve the equation, we can start by simplifying it further: 7r^2 = 36πr.
Dividing both sides of the equation by r, we get 7r = 36π.
Finally, dividing both sides of the equation by 7, we find that r = (36π)/7 cm.
Therefore, the radius of the cone is (36π)/7 cm.
Step 6: Find the vertical angle of the cone.
The vertical angle (V) of the cone can be determined using trigonometry. We can use the relationship between the slant height (ℓ) and the radius (r) of the cone, where tan(V) = r/ℓ.
Substituting the values of r = (36π)/7 cm and ℓ = 4π cm, we get tan(V) = [(36π)/7] / (4π).
Simplifying further, we have tan(V) = (9/7) / 4.
Using a calculator, we can take the inverse tangent (arctan) of both sides to find V.
Therefore, V = arctan((9/7) / 4).
Calculating this, we find that V ≈ 22.46 degrees.
Hence, the radius of the cone is approximately (36π)/7 cm and its vertical angle is approximately 22.46 degrees.