To find the volume of the cone formed when this sector is rolled up, let’s go through the steps carefully.

Step 1: Determine the Radius of the Base of the Cone

The given sector has a radius of 8 inches and a central angle of 80°. When rolled into a cone, the arc length of the sector becomes the circumference of the base of the cone.

1. Calculate the arc length of the sector:

   $\text{Arc length} = \frac{\theta}{360^\circ} \times 2 \pi r$ where:

   • $\theta = 80^\circ$
   • $r = 8$ inches

   $\text{Arc length} = \frac{80}{360} \times 2 \pi \times 8 = \frac{2}{9} \times 2 \pi \times 8$ $= \frac{16 \pi}{9} \approx 5.59 \text{ inches}$

   So, the circumference of the cone's base is approximately 5.59 inches.

2. Calculate the radius of the base of the cone:

   Since the circumference $C$ of the base is given by $C = 2 \pi R$, where $R$ is the radius of the base of the cone:

   $R = \frac{C}{2 \pi} = \frac{5.59}{2 \pi} \approx 0.89 \text{ inches}$

Step 2: Determine the Slant Height of the Cone

The slant height of the cone is the same as the radius of the original sector, which is 8 inches.

Step 3: Find the Height of the Cone

Using the Pythagorean theorem in the right triangle formed by the height $h$, radius $R$, and slant height $s$ (where $s = 8$ inches):

$h = \sqrt{s^2 - R^2} = \sqrt{8^2 - 0.89^2}$ $h = \sqrt{64 - 0.7921} \approx \sqrt{63.2079} \approx 7.95 \text{ inches}$

Step 4: Calculate the Volume of the Cone

The volume $V$ of a cone is given by:

$V = \frac{1}{3} \pi R^2 h$

Substituting $R \approx 0.89$ inches and $h \approx 7.95$ inches:

$V = \frac{1}{3} \pi (0.89)^2 \times 7.95$ $V \approx \frac{1}{3} \pi \times 0.7921 \times 7.95$ $V \approx \frac{1}{3} \pi \times 6.3$ $V \approx 6.6 \text{ cubic inches (to one decimal place)}$

Answer:

The volume of the cone is approximately 6.6 cubic inches.

Would you like further details on any step, or have any questions?

Here are five additional questions to expand your understanding:

1. How would the volume change if the central angle were different, say 120°?
2. What effect does increasing the radius of the sector have on the cone’s dimensions?
3. How do we calculate the surface area of this cone after it’s formed?
4. If the cone is filled with a liquid, how much liquid would it hold in liters?
5. What are other practical applications of converting sectors into cones?

Tip: Always double-check units when solving geometry problems, as volume and area formulas are sensitive to consistent units.