a sector with an angle 110 degree at the centre of a circle is cut away from a circular piece of paper of radius 70cm,the remaining part is folded to form a cone, find:
a:vertical angle of the cone
b:angle of the sector
the base radius of the cone is 250/360 * 70 = 48.6 cm
(a) sinθ = 48.6/70
(b) ? You just said it was 110°
a: Well, that's a "coneundrum!" To find the vertical angle of the cone, we need to look at the properties of circles and cones. Since the sector has an angle of 110 degrees at the center, it means that the remaining part of the paper also has an angle of 110 degrees at the top vertex of the cone. However, in a cone, the vertical angle is twice the angle at the top vertex. So, the vertical angle of the cone will be 2 * 110 = 220 degrees.
b: Now, let's tackle the angle of the sector. Since the sector is cut away from a circular piece of paper, it will have the same angle as at the center of the circle. So, the angle of the sector is 110 degrees.
Remember, it's important to always "circle" back to the given information and use a "cone-textual" understanding to solve the problem!
To find the vertical angle of the cone, we need to use the fact that the remaining part of the paper is folded into a cone.
a) Vertical angle of the cone:
The vertical angle of a cone is the angle formed by the slant height and the radius of the cone.
To find the slant height of the cone, we can use the Pythagorean theorem.
In a right triangle formed by the slant height (l), the radius (r), and the height (h) of the cone, we have:
l^2 = r^2 + h^2
Since we are given the radius of the circular piece of paper as 70 cm, and we know that the sector has an angle of 110 degrees at the center of the circle, we can find the height of the cone.
The arc length of the sector is given by (110 degrees/360 degrees) × 2π × r.
Arc length = (110/360) × 2π × 70
Let's calculate the arc length:
To find the vertical angle of the cone, we need to use the concept of similar triangles.
a) Vertical Angle of the Cone:
When a sector is cut from a circular piece of paper and folded to form a cone, the angle of the sector becomes the vertical angle of the cone. Therefore, the vertical angle of the cone is 110 degrees.
b) Angle of the Sector:
To determine the angle of the sector, we need to use the formula for the length of an arc in a circle. The formula is given by:
Arc Length = (Angle / 360) x (2πr)
In this case, the arc length is equal to the circumference of the circular piece of paper, minus the length of the arc cut out to form the sector. The circumference of the circular piece can be calculated by:
Circumference = 2πr
Given that the radius of the circular piece of paper is 70 cm, we can calculate the circumference:
Circumference = 2π × 70 cm
Next, we find the length of the arc cut out to form the sector. We know that the angle at the center is 110 degrees, and we need to convert it to radians:
Angle in Radians = (Angle in Degrees x π) / 180
Angle in Radians = (110 x π) / 180
Once we have the angle in radians, we can calculate the length of the arc using the formula:
Arc Length = Radius x Angle in Radians
Arc Length = 70 cm × (110π / 180)
Now, we can find the angle of the sector:
Angle of the Sector = (Arc Length / Circumference) x 360
Angle of the Sector = [(70 × 110π / 180) / (2π × 70)] x 360
Simplifying the equation, we find:
Angle of the Sector = 110 degrees
Thus, the angle of the sector is also 110 degrees.