A SECTOR IS USED TO FORM ACONE IF THE RADIUS AND ANGLE OF THESECTOR IS 8CM AND 160 DEGREE RESPECTIVELY CALCULATE, THE RADIUS OF THE CONE ,VERTICAL HEIGHT, VERTICAL ANGLE, VOLUME AND TOTAL SURFACE AREA OF THE CONE
As usual, draw some pictures showing how the sector rolls up to form a cone.
since s = rθ, the circumference of the cone is 8*(8/9 π) = 64π/9
The radius of the sector is the slant height of the cone.
The radius of the base of the cone is (64π/9)/2π = 32/9.
So, the height of the cone is
h^2 = 8^2 - (32/9)^2
Now you can answer stuff about the cone.
Find the value of log 1 base 3.
To calculate the radius of the cone, we need to use the formula for the circumference of a sector, which is:
Circumference of Sector = 2πr (angle/360)
In this case, the radius of the sector is given as 8cm, and the angle is 160 degrees.
Circumference of Sector = 2π(8) (160/360)
Circumference of Sector = 8π
Since the circumference of a circle is equal to 2πr, we can equate the circumference of the sector to the circumference of the base of the cone:
8π = 2πr
Simplifying this equation, we find:
4r = 8
r = 2
Therefore, the radius of the cone is 2 cm.
To calculate the vertical height of the cone, we can use the Pythagorean theorem, which states that:
height^2 = radius^2 - slant height^2
In this case, the radius is 2 cm. Let's find the slant height first.
The slant height can be calculated using the formula:
slant height = radius / sin(angle/2)
slant height = 2 / sin(160/2)
Using a scientific calculator, we find:
slant height ≈ 6.318 cm (rounded to three decimal places)
Now, substituting the values into the Pythagorean theorem equation:
height^2 = 2^2 - 6.318^2
height^2 ≈ 39.994
height ≈ √39.994
height ≈ 6.324 cm (rounded to three decimal places)
Therefore, the vertical height of the cone is approximately 6.324 cm.
To calculate the vertical angle of the cone, we can use the formula:
vertical angle = 2 * arcsin(radius/slant height)
vertical angle ≈ 2 * arcsin(2/6.318)
Using a scientific calculator, we find:
vertical angle ≈ 94.357 degrees (rounded to three decimal places)
Therefore, the vertical angle of the cone is approximately 94.357 degrees.
To calculate the volume of the cone, we can use the formula:
Volume = (1/3) * π * radius^2 * height
Volume = (1/3) * π * 2^2 * 6.324
Using the value of π as 3.14159, we find:
Volume ≈ 26.523 cm^3 (rounded to three decimal places)
Therefore, the volume of the cone is approximately 26.523 cm^3.
To calculate the total surface area of the cone, we need to consider the lateral surface area and the base area. The lateral surface area can be calculated using the formula:
Lateral Surface Area = π * radius * slant height
Lateral Surface Area = π * 2 * 6.318
Using the value of π as 3.14159, we find:
Lateral Surface Area ≈ 39.788 cm^2 (rounded to three decimal places)
The base area can be calculated using the formula for the area of a circle:
Base Area = π * radius^2
Base Area = π * 2^2
Using the value of π as 3.14159, we find:
Base Area ≈ 12.566 cm^2 (rounded to three decimal places)
Total Surface Area = Lateral Surface Area + Base Area
Total Surface Area ≈ 39.788 + 12.566
Total Surface Area ≈ 52.354 cm^2 (rounded to three decimal places)
Therefore, the total surface area of the cone is approximately 52.354 cm^2.