A sector of radius 7cm substending an angle 210 at the center of the circle, is used to form a cone calculate to the nearest whole number,the base radius of the cone,the height of the cone, the total surface arc of the cone, volume of the cone, vertical angle of the cone.
Mathematics.
What about the volume of the cone
YES
What is the Volume of the cone
V=1/3×22/7×16×36=101cm
I'm sorry, but that's not correct. The correct formula for the volume of a cone is V = 1/3 * π * r^2 * h, where r is the base radius and h is the height of the cone.
Plugging in the values we have calculated, we get:
V = 1/3 * π * (49/12)^2 * (5.686)
V ≈ 77.2 cm^3
Therefore, the volume of the cone is approximately 77.2 cm^3.
What is then the vertical angle of the cone
To find the vertical angle of the cone, we need to use the formula:
tan(angle) = (opposite side) / (adjacent side)
In this case, the opposite side is the height of the cone (h) and the adjacent side is the radius of the base (r). So, we have:
tan(angle) = h/r
tan(angle) = (5.686) / (49/12)
tan(angle) ≈ 0.734
Now, we need to take the inverse tangent (tan^-1) of 0.734 to find the angle. Using a calculator, we get:
angle ≈ 36.6 degrees
Therefore, the vertical angle of the cone is approximately 36.6 degrees.
To calculate the base radius of the cone, we need to find the length of the arc that makes up the sector.
The circumference of a full circle is given by the formula C = 2πr, where r is the radius of the circle.
Since the sector has an angle of 210 degrees (or 210/360 = 7/12 in terms of the full circle), the length of the arc that makes up the sector is (7/12) * C.
Given that the radius of the sector is 7cm, we can substitute this into the formula to find the length of the arc:
Arc Length = (7/12) * 2π * 7
= (7/12) * 14π
= 7π/6
Now let's move on to calculating the base radius of the cone. The base of the cone is formed by the circumference of the sector, which is the same as the length of the arc.
Base Radius of Cone = Arc Length / (2π)
= (7π/6) / (2π)
= 7/12
≈ 0.58 cm (to the nearest whole number)
Next, let's calculate the height of the cone. We can use the Pythagorean theorem, where the height (h) is the perpendicular distance from the apex of the cone to the base.
In the sector, the radius forms the hypotenuse of a right-angled triangle, and the height is one of the legs. The other leg is the base radius of the cone we just calculated.
Using the Pythagorean theorem:
(7 cm)^2 = (base radius)^2 + (height)^2
49 = (7/12)^2 + (height)^2
49 = 49/144 + (height)^2
49 = 49/144 + (height)^2
48/144 = (height)^2
1/3 = (height)^2
height = √(1/3)
height ≈ 0.58 cm (to the nearest whole number)
Now let's find the total surface area of the cone. It consists of the curved surface area of the cone (lateral surface area) and the base area.
Curved Surface Area (Lateral Area) of Cone = π * base radius * slant height
The slant height can be found by applying trigonometry to the cone. Since the angle at the center of the sector is 210 degrees, the angle at the apex of the cone is half of that, which is 210/2 = 105 degrees.
Using the formula: slant height = radius / sin(angle)
slant height = 7 / sin(105°)
The base area of the cone is given by: base area = π * base radius^2
Total Surface Area = Curved Surface Area + Base Area
= π * base radius * slant height + π * base radius^2
= π * (7/12) * (7/sin(105°)) + π * (7/12)^2
Now let's calculate the volume of the cone. The formula for the volume of a cone is:
Volume of Cone = (1/3) * base area * height
Substituting the values we have found:
Volume of Cone = (1/3) * π * (7/12)^2 * √(1/3)
Finally, let's calculate the vertical angle of the cone. The vertical angle is the angle formed by the apex of the cone and the edge along the base.
The vertical angle can be found using the formula:
Vertical Angle = 2 * arctan(height / base radius)
Substituting the values:
Vertical Angle = 2 * arctan(0.58 / 0.58)
= 2 * arctan(1)
= 2 * π/4
= π/2 radians
After performing the calculations, we get the following results (rounded to the nearest whole number):
Base radius of the cone ≈ 1 cm
Height of the cone ≈ 1 cm
Total surface area of the cone ≈ 9 cm²
Volume of the cone ≈ 1 cm³
Vertical angle of the cone ≈ 1 radian
The arc length of the sector becomes the circumference of the base of the cone.
arc = (210)/(360)*14π = 49/6 π cm
let the radius of the circular base be r
2πr = 49/6π
r = 49/12 cm
We can now find the height of the cone:
h^2 + (49/12)^2 = 7^2
h^2 = 4655/144
h = √(4655/144) = appr 5.686
Now you have all the needed data.
Use the formulas given to you for volume etc.