A 210 degrees of a sector of radius 5cm is bent on form cone. For the fone formed find a.the base radius of the cone b.the vertical angle of the cone c.the curved surface area of the cone.
If you make a sketch, you will see that the arc length subtended by the 210° sector angle will become the circumference of the base of the cone.
arc/(10π) = 210/360
arc = 10π(1/3) = 10π/3
let the radius of the base be r
2πr = 10π/3
r = 10π/(6π) = 5/3
the slant height of the cone = 5
tanØ = 5/(5/3) = 1/3, where Ø is the base angle
I will let you find the angle, as well as the surface area of the cone.
for b) how does the curved surface of the cone related to the area of the sector??
To solve this problem, we will first find the circumference of the whole circle based on the given sector angle. Then, we will use this circumference to find the base radius and the curved surface area of the cone. Finally, we will calculate the vertical angle of the cone using the given radius of the sector.
Given:
- Sector angle = 210 degrees
- Radius of the sector = 5 cm
Step 1: Calculate the circumference of the whole circle
The sector angle is given as 210 degrees out of 360 degrees, so the fraction of the circle represented by the sector is:
Fraction of the circle = (210 degrees) / (360 degrees) = 7/12
The circumference (C) of a full circle with the sector radius is given by the formula:
C = 2πr, where r is the radius
In this case, the given radius is 5 cm. Therefore:
C = 2π(5) = 10π
Step 2: Calculate the base radius of the cone (a)
The circumference of the sector represents the base circumference of the cone. The base circumference (C_b) of a cone is given by the formula:
C_b = 2πr_b, where r_b is the base radius of the cone
We know that the base circumference of the cone is 10π, so we can set up the equation:
10π = 2πr_b
Dividing both sides by 2π, we get:
r_b = 5 cm
Therefore, the base radius of the cone is 5 cm.
Step 3: Calculate the vertical angle of the cone (b)
The vertical angle (θ) of the cone can be found using the radius of the sector (r) and the height of the cone (h) with the help of the Pythagorean theorem:
r^2 + h^2 = l^2, where l is the slant height of the cone
In this case, the given radius of the sector is 5 cm. Since the slant height is the same as the radius of the sector (as it forms a lateral surface of the cone), we have:
5^2 + h^2 = 5^2
Simplifying the equation, we get:
h^2 = 0
This means that the height of the cone is 0, which is not possible. Therefore, there is an error in the given information or the given sector angle.
Step 4: Calculate the curved surface area of the cone (c)
The curved surface area (A) of a cone is given by the formula:
A = πrl, where r is the base radius of the cone and l is the slant height of the cone
Since we couldn't calculate the slant height in Step 3 due to the error in the information given, we cannot accurately determine the curved surface area of the cone.
In summary:
a. The base radius of the cone is 5 cm.
b. The vertical angle of the cone cannot be determined due to insufficient information or an error.
c. The curved surface area of the cone cannot be determined due to insufficient information or an error.
To find the base radius of the cone, you need to use the given information about the sector of the circle.
a) The arc length of the sector is given as 210 degrees of a circle with a radius of 5 cm. To find the arc length, you can use the formula:
Arc Length = (θ/360) * (2πr),
where θ is the angle in degrees, r is the radius of the circle, and π is a constant value approximately equal to 3.14.
Plugging in the values, we have:
Arc Length = (210/360) * (2 * 3.14 * 5)
= (7/12) * (31.4)
= 18.2 cm (rounded to one decimal place)
The arc length of the sector, which will be the circumference of the base of the cone, is 18.2 cm.
Since the circumference of a circle is given by the formula:
Circumference = 2πr,
you can find the base radius (r) of the cone by rearranging the formula:
r = Circumference / (2π).
Plugging in the value of the arc length (Circumference), we get:
r = 18.2 / (2 * 3.14)
= 2.9 cm (rounded to one decimal place).
Therefore, the base radius of the cone is approximately 2.9 cm.
b) To find the vertical angle of the cone, you can use the fact that the angle of the sector is also equal to the angle at the vertex of the cone. In this case, the angle is given as 210 degrees.
Thus, the vertical angle of the cone is 210 degrees.
c) The curved surface area of the cone can be found using the formula:
Curved Surface Area = π * r * l,
where r is the radius of the cone's base and l is the slant height of the cone.
To find the slant height, you can use the formula:
l = sqrt(r^2 + h^2),
where h is the height of the cone. Since the height is not given, we need to calculate it.
The arc length of the sector is given as 18.2 cm, which represents the circumference of the base of the cone. So, the circumference can be found using the formula:
Circumference = 2πr.
Plugging in the value of the radius (2.9 cm), we have:
18.2 = 2 * 3.14 * 2.9,
r = 5 cm.
Now, we can calculate the height of the cone using the formula:
h = (Arc Length / 360) * (2πr).
Plugging in the values, we get:
h = (18.2 / 360) * (2 * 3.14 * 5)
≈ 0.159 * 31.4
≈ 4.97 cm (rounded to two decimal places).
Therefore, the height of the cone is approximately 4.97 cm.
Now, we can calculate the slant height of the cone:
l = sqrt(r^2 + h^2)
= sqrt(5^2 + 4.97^2)
= sqrt(25 + 24.7)
= sqrt(49.7)
≈ 7.05 cm (rounded to two decimal places).
Finally, we can find the curved surface area of the cone:
Curved Surface Area = π * r * l
= 3.14 * 2.9 * 7.05
≈ 58.99 cm^2 (rounded to two decimal places).
Therefore, the curved surface area of the cone is approximately 58.99 cm^2.