Dana takes a sheet of paper, cuts a 120-degree 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.

The circumference of the whole circle from which you cut the sector is 2π(12) = 24π cm

so using ratios to find the circumference of the sector:
120/360 = x/24π
x = 8π

That becomes the circumference of the base of the cone
for its radius r :
2πr = 8π
r = 4 cm
The radius of the original sector becomes the slant side of the cone
so h^2 + 4^2 = 12^2
h = √128 = 8√2

vol = πr^2h/3 = π(4)^2(8√2)/3 = (128π√2)/3 cm^3 or appr. 189.563

Well, Dana seems to be quite the paper artist! Let's see what we can do with this.

To find the height of the cone, we need to first find the slant height. Since the angle of the sector is 120 degrees and the radius of the sector is 12 cm, the length of the arc of the sector would be (120/360) * 2π(12 cm) = 8π cm.

Now, the slant height is the hypotenuse of the right triangle formed by the sector and the height of the cone. We can use the Pythagorean theorem to find the slant height.

Let's call the height of the cone h.

Using the Pythagorean theorem, we have:
(slant height)^2 = (radius)^2 + (height)^2
(slant height)^2 = (12 cm)^2 + h^2
(slant height)^2 = 144 cm^2 + h^2

Since the slant height is 8π cm, we can substitute that in:
(8π cm)^2 = 144 cm^2 + h^2
64π^2 cm^2 = 144 cm^2 + h^2

Now, let's solve for h:
h^2 = 64π^2 cm^2 - 144 cm^2
h^2 = (64π^2 - 144) cm^2
h = √(64π^2 - 144) cm

Now, let's find the volume of the cone. The 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.

In this case, the radius of the circular base is 12 cm, and we have found the height to be √(64π^2 - 144) cm. So, we can substitute these values into the formula:

V = (1/3)π(12 cm)^2(√(64π^2 - 144) cm)
V = (1/3)π(144 cm^2)(√(64π^2 - 144) cm)
V = (48π cm^3)(√(64π^2 - 144) cm)
V ≈ (48π)(8.127 cm)
V ≈ 392.71π cm^3

So, the height of the cone is approximately √(64π^2 - 144) cm, and the volume of the paper cone is approximately 392.71π cm^3.

To find the height and volume of the paper cone, we can follow these steps:

Step 1: Finding the circumference of the base of the cone:
The circumference of the base of the cone is equal to the length of the 120-degree circular sector which Dana cut out.

To find the circumference, we can use the formula for the circumference of a circle:

C = 2πr

Where C is the circumference and r is the radius.

Given that the radius is 12 cm, we can substitute the value into the formula:

C = 2π(12)
C = 24π cm

Now we have the circumference of the base of the cone, which is 24π cm.

Step 2: Finding the height of the cone:
To find the height of the cone, we need to know the slant height of the cone. The slant height is the distance from the tip of the cone to the edge of the circular sector.

Since the circular sector's central angle is 120 degrees, and the sector radius is 12 cm, we can use the trigonometric ratio sine to find the slant height.

The formula for the slant height is:

l = r / sinθ

Where l is the slant height, r is the radius, and θ is the central angle in radians.

To convert degrees to radians, we can use the formula:

θ (in radians) = θ (in degrees) * (π / 180)

Substituting the values into the formula, we have:

θ = 120 * (π / 180)
θ = 2π / 3 radians

Using this value for θ and the radius r = 12 cm, we can find the slant height:

l = 12 / sin(2π / 3)
l ≈ 12 / 0.866
l ≈ 13.86 cm

So the slant height of the cone is approximately 13.86 cm.

To find the height, we can use the Pythagorean theorem:

h² = l² - r²

Substituting the values we have:

h² = (13.86)² - (12)²
h² ≈ 192.0996 - 144
h² ≈ 48.0996
h ≈ √48.0996
h ≈ 6.93 cm

Therefore, the height of the cone is approximately 6.93 cm.

Step 3: Finding the volume of the cone:
The volume of a cone can be found using the formula:

V = (1/3)πr²h

Substituting the given values:

V = (1/3)π(12)²(6.93)
V ≈ (1/3)π(144)(6.93)
V ≈ (1/3)(451.848)
V ≈ 150.616 cm³

Therefore, the volume of the paper cone is approximately 150.616 cm³.

To find the height and volume of this paper cone, we can follow these steps:

1. Find the circumference of the base of the cone.
- The circumference of a circle is calculated using the formula: C = 2πr, where r is the radius.
- In this case, the base of the cone is a circle with a radius of 12 cm, so the circumference is C = 2π(12) cm.

2. Determine the slant height of the cone.
- The slant height is the distance from the tip of the cone to any point on the circumference of the base. It can be found using trigonometry.
- In this case, the sector angle is 120 degrees, which means the central angle of the sector is 360 - 120 = 240 degrees (since the sum of all angles in a circle is 360 degrees).
- The radius forms the hypotenuse, and the slant height forms the opposite side of a right triangle within the sector.
- We can use the sine function to find the slant height: sin(angle) = opposite / hypotenuse.
- sin(240 degrees) = slant height / 12 cm.
- Rearranging the formula, we have: slant height = 12 cm * sin(240 degrees).

3. Compute the height of the cone.
- The height is the perpendicular distance from the tip of the cone to the base, which can be found using the Pythagorean theorem.
- Using the slant height we just calculated, and the radius (12 cm), we can find the height (h) using the formula: h = sqrt((slant height)^2 - (radius)^2).
- Substitute the values into the formula: h = sqrt((slant height)^2 - (12 cm)^2).

4. Calculate the volume of the cone.
- The volume of a cone can be found using the formula: V = (1/3)πr^2h, where r is the radius and h is the height.
- Substitute the values into the formula: V = (1/3)π(12 cm)^2 * h.

Using these steps, you can compute the height and volume of the paper cone.