Question

Dana takes a sheet of paper, cuts a 120° circular sector from it, then rolls it up and tapes the straight edges together to form a cone. Given that the sector radius is 12 cm, find the height and volume of this paper cone.

Answer 1

To find the height and volume of the paper cone, calculate the slant height using the radius and central angle, then find the height by subtracting the radius from the slant height. Finally, calculate the volume using the height and radius.

Step-by-step explanation:

To find the height and volume of the paper cone, we first need to calculate the slant height of the sector. The slant height can be found using the radius and the central angle. Since the central angle is 120° and the radius is 12 cm, the slant height can be calculated as follows:

Slant height = 2 * radius * sin(angle/2)

Slant height = 2 * 12 cm * sin(120°/2)

Slant height = 24 cm * sin(60°)

Slant height = 24 cm * √(3/2)

Slant height ≈ 24 cm * 1.2247 ≈ 29.3947 cm

Next, we can find the height of the cone by using the slant height and the radius:

Height = slant height - radius

Height = 29.3947 cm - 12 cm

Height ≈ 17.3947 cm

Finally, we can calculate the volume of the cone using the radius and height:

Volume = (1/3) * π * r² * h

Volume = (1/3) * π * 12² * 17.3947

Volume ≈ 865.694 cm³

Answer 2

The height of the cone is approximately 18.39 cm and the volume is approximately 1,854.66 cm³.

To find the height of the cone, we need to calculate the circumference of the base of the cone. Since the circular sector has an angle of 120°, the length of the circumference is 120/360 times the circumference of a full circle.

So, the circumference of the base is (120/360) × 2π × 12 cm = 8π cm.

The circumference of the base is equal to the length of the slant height of the cone, which is the hypotenuse of a right triangle with height h and radius r.

Using the Pythagorean theorem, we can write the equation:

`8\pi cm = √((h^2 + 12^2)) cm`.

By squaring both sides of the equation, we get:

`(8\pi)^2 cm^2 = h^2 + 12^2 cm^2`.

Solving for h, we find:

`h = √([(8\pi)^2 - 12^2]) cm`.

Using a calculator, we get h ≈ 18.39 cm.

The volume of the cone can be calculated using the formula
`V = (1/3)\pi r^2h`.

Plugging in the values, we get:

`V = (1/3)\pi(12 cm)^2(18.39 cm) \approx 1,854.66 cm^3`.