To solve this problem, we need to calculate the volume of a cone and then subtract the volume of a bubble gum from it. Here are the detailed steps:
1. Calculate the volume of the cone:
- The formula for the volume of a cone is [tex]\( V = \frac{1}{3} \pi r^2 h \)[/tex]
- Given:
- Radius of the cone, [tex]\( r = 4 \)[/tex] inches
- Height of the cone, [tex]\( h = 6 \)[/tex] inches
- [tex]\( \pi = 3.14 \)[/tex]
- Plug the values into the formula:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \times 3.14 \times 4^2 \times 6
\][/tex]
2. Calculate the volume of the bubble gum:
- The formula for the volume of a sphere is [tex]\( V = \frac{4}{3} \pi r^3 \)[/tex]
- Given:
- Diameter of the bubble gum, [tex]\( d = 0.8 \)[/tex] inches, so the radius, [tex]\( r = \frac{0.8}{2} = 0.4 \)[/tex] inches
- [tex]\( \pi = 3.14 \)[/tex]
- Plug the values into the formula:
[tex]\[
V_{\text{bubble gum}} = \frac{4}{3} \times 3.14 \times 0.4^3
\][/tex]
3. Subtract the volume of the bubble gum from the volume of the cone:
- The volume of the part of the cone that can be filled with flavored ice is:
[tex]\[
V_{\text{ice}} = V_{\text{cone}} - V_{\text{bubble gum}}
\][/tex]
Now, let's identify the corresponding correct formula from the given options. We'll match the mathematical expressions:
- Option 1: [tex]\(\frac{1}{3}(3.14)\left(6^2\right)(4)-\frac{4}{3}(3.14)\left(0.4^3\right)\)[/tex]
- Option 2: [tex]\(\frac{1}{3}(3.14)\left(4^2\right)(6)-\frac{4}{3}(3.14)\left(0.4^3\right)\)[/tex]
- Option 3: [tex]\(\frac{1}{3}(3.14)\left(6^2\right)(4)-\frac{4}{3}(3.14)\left(0.8^3\right)\)[/tex]
- Option 4: [tex]\(\frac{1}{3}(3.14)\left(4^2\right)(6)-\frac{4}{3}(3.14)\left(0.8^3\right)\)[/tex]
For the correct calculation:
- [tex]\( V_{\text{cone}} = \frac{1}{3} (3.14)(4^2)(6) \)[/tex]
- [tex]\( V_{\text{bubble gum}} = \frac{4}{3} (3.14)(0.4^3) \)[/tex]
Comparing it with the options, Option 2 matches our derived formulas:
[tex]\[
\frac{1}{3}(3.14)\left(4^2\right)(6)-\frac{4}{3}(3.14)\left(0.4^3\right)
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{\frac{1}{3}(3.14)\left(4^2\right)(6)-\frac{4}{3}(3.14)\left(0.4^3\right)}
\][/tex]
