The sector shown is formed into a cone by joining its two radii. Calculate the volume of the cone correct to the nearest whole number.
It shows a diagram with a 240 degree sector or two-thirds of a circle with a radius of 12 cm.
How do I do this?
Thank you so much
Sometimes it helps to have an actual model of your problem.
Take a piece of paper and cut out a sector approximating your given sector.
Form it into a cone.
You will see that the radius of the sector becomes the "slant height" of the cone, and the arc length of the sector becomes the circumference of the base of the cone.
So, let's find the radius of the cone:
The arc length of our sector would be 240/360 of the circumference of the circle from which the sector was obtained.
so (240/36)( 24π) = 16π
then for the base of the cone
2πr = 16π
r = 8
Now we can find the height of the cone:
h^2 + 8^2 = 12^2
h = √80
Volume of cone = (1/3) base x height
= (1/3)(64π)(√80)
= appr. 599 to the nearest whole number
Well, I must say this question is really trying to shape things up! So, to calculate the volume of the cone formed by this sector, we first need to find the radius of the cone. Since the sector is two-thirds of a circle, we can use proportion to find the radius.
We know that the length of the sector is 240 degrees out of 360 degrees, so we can set up a proportion:
(240 degrees / 360 degrees) = (12 cm / r)
Now, let's solve for r!
240/360 = 12/r
Cross-multiplying, we have:
240r = 360 * 12
Dividing both sides by 240:
r = (360 * 12) / 240
r = 18 cm
Now that we have the radius, we can calculate the volume of the cone using the formula:
Volume = (1/3) * π * r^2 * h
In this case, since we joined the radii to form the cone, the height of the cone will also be 12 cm.
Volume = (1/3) * π * 18^2 * 12
Volume ≈ 11089 cm³
Rounding to the nearest whole number, the volume of the cone is approximately 11089 cm³.
I hope this answer cones in handy!
To find the volume of the cone formed by joining the two radii, you can follow these steps:
1. Calculate the angle in radians of the sector by converting the given angle in degrees to radians. Since there are 360 degrees in a circle, and the sector is 240 degrees, we can use the formula: angle in radians = (angle in degrees) x (π/180). Therefore, the angle in radians is (240 degrees) x (π/180) ≈ 4.19 radians.
2. Calculate the height of the cone using the radius and the angle in radians. Since the sector is formed by two radii, the length of the arc forming the sector is equal to the circumference of the base of the cone. The length of the arc is given by the formula: arc length = (angle in radians) x (radius). Therefore, the length of the arc is (4.19 radians) x (12 cm) ≈ 50.28 cm.
3. The height of the cone is given by the length of the arc divided by the circumference of the base, which is equal to the sum of the two radii. Therefore, the height of the cone is 50.28 cm / (2π(12 cm)) ≈ 0.66 cm.
4. Finally, we can calculate the volume V of the cone using the formula: V = (1/3) x base area x height, where the base area is given by the formula: base area = π(radius^2). Therefore, the volume of the cone is (1/3) x π x (12 cm)^2 x 0.66 cm ≈ 100 cm^3.
So, the volume of the cone, to the nearest whole number, is approximately 100 cubic centimeters.
To calculate the volume of the cone formed by joining the two radii of the sector, we need to follow a few steps:
1. First, we need to find the radius of the cone. Since the radius of the sector is given as 12 cm, we can use this value as the radius of the cone.
2. Next, we need to find the height of the cone. To do this, we need to find the length of the arc of the sector. The formula for the arc length is given by:
arc length = (central angle/360 degrees) * circumference of the circle
In this case, the central angle of the sector is given as 240 degrees, and the circumference of the circle is given by 2 * π * radius, where π is approximately 3.14159. Substituting these values into the formula, we can find the arc length.
arc length = (240/360) * (2 * π * 12)
= (2/3) * (2 * 3.14159 * 12)
= (2/3) * (2 * 3.14159 * 12)
≈ 25.13272 cm
The arc length is approximately 25.13272 cm.
Now, the height of the cone is equal to the radius of the sector, which is given as 12 cm.
3. Finally, using the formula for the volume of a cone, which is given by:
volume = (1/3) * π * radius^2 * height
Substituting the values we have obtained, we can calculate the volume of the cone:
volume = (1/3) * 3.14159 * 12^2 * 12
= (1/3) * 3.14159 * 144 * 12
= 3.14159 * 1728
≈ 5416
Therefore, the volume of the cone, rounded to the nearest whole number, is approximately 5416 cubic centimeters.