A 160 degree of a sector radius 10.5cm is bent to form a cone,find the vertical angle of the cone
make a sketch to see that the 10.5 cm becomes the hypotenuse of the cone and the circumference of the cone is the arc length of the sector.
160° = 160/180π radians
= 2.7925.. radians
arc = 2.7925(10.5) = 29.32153..
this becomes the circumf of the base of the cone
2πr = 2932153
r = 4.666... = 4 2/3 = 14/3
If 2Ø is the vertical angle, then
sinØ = 4.6666.../10.5
Ø = 26.387..
2Ø = 52.78°
We all that sine is equal to opposite over hypotenuse.If you sketch your diagram.The opposite will be 10.5cm while the 160°will be your hypotenuse. Area of a sector of a circle=160/360=0.444. Sine theta=160/360.sine theta=0.444. inverse of sine gives you 26.356. Vertical angle=2*theta will give 2*26.356=52.718.Then your vertical angle will be 53°(to the nearest degree).
To find the vertical angle of the cone formed, we can use the equation:
Vertical angle = 360° - Central angle of sector
Given that the central angle of the sector is 160°, we can substitute this value into the equation:
Vertical angle = 360° - 160°
Vertical angle = 200°
Therefore, the vertical angle of the cone is 200°.
To find the vertical angle of the cone, we need to use the concept of sector and relate it to a cone.
Given:
Angle of the sector (θ) = 160 degrees
Radius of the sector (r) = 10.5 cm
Step 1: Find the circumference of the sector
The circumference of the sector is equal to the circumference of the base of the cone formed.
Circumference of the sector = 2πr
Circumference of the sector = 2 × 3.14 × 10.5 cm
Circumference of the sector ≈ 65.97 cm
Step 2: Relate the circumference of the sector to the circumference of the cone
The circumference of the cone is equal to the circumference of the base of the cone formed.
Circumference of the cone = 2πr
Step 3: Calculate the radius of the cone
Since the angle of the sector is bent to form a cone, we can assume that the sector undergoes a complete revolution. So, the circumference of the cone is also equal to the circumference of the sector.
So, 2πr = 65.97 cm
Step 4: Calculate the radius of the cone
2πr = 65.97 cm
2 × 3.14 × r = 65.97 cm
6.28r = 65.97 cm
r ≈ 10.49 cm
Step 5: Find the slant height of the cone
The slant height (l) of the cone can be found using the Pythagorean theorem.
l^2 = r^2 + h^2
Step 6: Find the height of the cone
Since the given angle is the vertical angle of the cone, we can use it to calculate the height.
For a cone, the vertical angle (θ) is related to the slant height (l) and the radius (r) by the following equation:
sin(θ/2) = r / l
Plugging in the values:
sin(θ/2) = 10.49 / l
Solving for l:
l = 10.49 / sin(θ/2)
l = 10.49 / sin(160/2)
l ≈ 10.68 cm
Step 7: Calculate the height of the cone
Using the Pythagorean theorem:
l^2 = r^2 + h^2
Substituting the known values:
(10.68)^2 = (10.49)^2 + h^2
Solving for h:
h^2 = (10.68)^2 - (10.49)^2
h ≈ 1.64 cm
Step 8: Find the vertical angle of the cone
The vertical angle of the cone can be calculated using the following formula:
Vertical angle (θ) = 2 × arcsin(h / l)
Vertical angle (θ) = 2 × arcsin(1.64 / 10.68)
Vertical angle (θ) ≈ 17.42 degrees
Therefore, the vertical angle of the cone is approximately 17.42 degrees.