1) Find,in the term π the curved surface area of cone with circular base diameter 10cm and height 12cm
2)If the cone in question 1 is made of paper and the paper is flattened out into the sector of a circle, what is the angle of the sector
1) "in terms of π" -- I know this is math, not English, but it helps if your question actually makes sense. Now recall that
A = πrl
l^2 = r^2 + h^2
Now plug in your numbers
2) the radius of the sector is the slant height of the cone (see l, above)
the arc length is the circumference of the base of the cone: πd
since s = rθ for a sector, you just need to solve
lθ = πd
so θ=πd/l
1) To find the curved surface area of a cone, we can use the formula:
CSA = π * r * l
where "CSA" is the curved surface area, "π" is a constant approximately equal to 3.14159, "r" is the radius of the circular base, and "l" is the slant height of the cone.
Given that the diameter of the circular base is 10 cm, we can find the radius by dividing it by 2:
radius (r) = diameter / 2 = 10 cm / 2 = 5 cm
To find the slant height (l), we can use the Pythagorean theorem. The slant height, height, and radius form a right-angled triangle. The height is given as 12 cm. Therefore, we can calculate the slant height as follows:
l = √(h² + r²)
l = √(12 cm)² + (5 cm)²
l = √(144 cm² + 25 cm²)
l = √(169 cm²)
l = 13 cm
Now, we can calculate the curved surface area (CSA):
CSA = π * r * l
CSA = 3.14159 * 5 cm * 13 cm
CSA ≈ 201.0619 cm²
Therefore, the curved surface area of the cone is approximately 201.0619 cm².
2) To find the angle of the sector formed when the cone is flattened out, we can use the formula:
angle = (CSA / π * r²) * 360 degrees
where "angle" is the angle of the sector, "CSA" is the curved surface area, "π" is a constant approximately equal to 3.14159, and "r" is the radius of the circular base.
From question 1, we know that the curved surface area of the cone is approximately 201.0619 cm², and the radius of the circular base is 5 cm.
angle = (201.0619 cm² / 3.14159 * 5 cm²) * 360 degrees
angle = (201.0619 cm² / 78.53975 cm²) * 360 degrees
angle = (2.5584) * 360 degrees
angle ≈ 920.6384 degrees
Therefore, the angle of the sector when the cone is flattened out is approximately 920.6384 degrees.
To find the curved surface area of a cone, we can use the formula:
CSA = πrℓ
where "CSA" represents the curved surface area, "r" is the radius of the circular base, and "ℓ" is the slant height of the cone.
1) In this case, we are given the diameter of the circular base, which is 10 cm. Since the radius is half the diameter, the radius (r) would be 10/2 = 5 cm.
Now, to find the slant height (ℓ) of the cone, we can use the Pythagorean theorem. The slant height is the hypotenuse of a right triangle formed by the height and the radius.
Using the Pythagorean theorem:
ℓ^2 = r^2 + h^2
Since we know the radius (r) is 5 cm and the height (h) is 12 cm, we can substitute these values into the equation:
ℓ^2 = 5^2 + 12^2
ℓ^2 = 25 + 144
ℓ^2 = 169
Taking the square root of both sides, we find:
ℓ = √169
ℓ = 13 cm
Now, we can substitute the values into the formula for curved surface area:
CSA = π(5)(13)
CSA = 65π square cm
Therefore, the curved surface area of the cone is 65π square cm.
2) If the cone is made of paper and flattened out into the sector of a circle, we need to find the angle of the sector.
The circumference of the base of the cone would be the same as the circumference of the sector. The circumference of a circle is given by the formula:
C = 2πr
In this case, the radius (r) is 5 cm (half the diameter). Thus, the circumference is:
C = 2π(5)
C = 10π cm
Since the sector is formed by the curved surface of the cone, the length of the arc of the sector would be the same as the curved surface area of the cone.
From question 1, we found that the curved surface area of the cone is 65π square cm.
Using the formula for the length of an arc of a circle:
L = (θ/360) * 2πr
where "L" is the length of the arc, "θ" is the angle (in degrees) of the sector, and "r" is the radius.
We can now substitute the known values:
65π = (θ/360) * 10π
Simplifying the equation yields:
65 = (θ/360) * 10
To isolate θ, we can cross-multiply and then solve for θ:
650 = 10θ
Dividing both sides by 10:
θ = 65 degrees
Therefore, the angle of the sector formed by flattening out the paper cone would be 65 degrees.