Let's break down the steps to determine which formula calculates the volume of water in the vase:
1. Calculate the volume of the vase:
- The vase is a cylinder with a diameter of 6 inches and a height of 12 inches.
- The radius of the vase is half of the diameter:
[tex]\[
r_{\text{vase}} = \frac{6 \text{ inches}}{2} = 3 \text{ inches}
\][/tex]
- The formula for the volume of a cylinder is [tex]\(V = \pi r^2 h\)[/tex], where [tex]\(r\)[/tex] is the radius and [tex]\(h\)[/tex] is the height.
- Substituting the radius and height:
[tex]\[
V_{\text{vase}} = \pi (3 \text{ inches})^2 (12 \text{ inches}) = \pi \times 9 \text{ in}^2 \times 12 \text{ in} = 108 \pi \text{ in}^3
\][/tex]
2. Calculate the volume of one marble:
- Each marble is a sphere with a diameter of 3 inches.
- The radius of each marble is half of the diameter:
[tex]\[
r_{\text{marble}} = \frac{3 \text{ inches}}{2} = 1.5 \text{ inches}
\][/tex]
- The formula for the volume of a sphere is [tex]\(V = \frac{4}{3} \pi r^3\)[/tex], where [tex]\(r\)[/tex] is the radius.
- Substituting the radius:
[tex]\[
V_{\text{marble}} = \frac{4}{3} \pi (1.5 \text{ inches})^3 = \frac{4}{3} \pi \times 3.375 \text{ in}^3 = 4.5 \pi \text{ in}^3
\][/tex]
3. Calculate the total volume of the 7 marbles:
[tex]\[
V_{\text{total marbles}} = 7 \times V_{\text{marble}} = 7 \times 4.5 \pi \text{ in}^3 = 31.5 \pi \text{ in}^3
\][/tex]
4. Calculate the volume of water in the vase:
- The volume of water is the volume of the vase minus the total volume of the marbles.
- Substituting the volumes:
[tex]\[
V_{\text{water}} = V_{\text{vase}} - V_{\text{total marbles}} = 108 \pi \text{ in}^3 - 31.5 \pi \text{ in}^3 = 76.5 \pi \text{ in}^3
\][/tex]
Now let's compare this to the provided options:
- First option: [tex]\(\pi(12 \text{ in})^2(3 \text{ in}) - 7\left(\frac{4}{3} \pi(1.5 \text{ in})^3\right)\)[/tex]
This doesn't match our expression for the vase volume calculation or marble volume correctly.
- Second option: [tex]\(\pi(3 \text{ in})^2(12 \text{ in}) - 7\left(\frac{4}{3} \pi(1.5 \text{ in})^3\right)\)[/tex]
This matches our calculations exactly:
- The volume of the vase [tex]\(\pi (3 \text{ in})^2 (12 \text{ in})\)[/tex]
- The total volume of the marbles [tex]\(7\left(\frac{4}{3} \pi (1.5 \text{ in})^3\right)\)[/tex]
This is the correct formula.
- Third option: [tex]\(n(12 \text{ in})^2(3 \text{ in}) - 1.5\left(\frac{4}{3} \pi(7 \text{ in})^2\right)\)[/tex]
This doesn't match our calculation at all.
- Fourth option: [tex]\(\pi(3 \text{ in})^2(12 \text{ in}) - 1.5\left(\frac{4}{3} \pi(7 \text{ in})^3\right)\)[/tex]
This also doesn't align with our values.
So, the correct formula is:
[tex]\[
\pi(3 \text{ in})^2(12 \text{ in})-7\left(\frac{4}{3} \pi(1.5 \text{ in})^3\right)
\][/tex]
