To determine which formula correctly calculates the volume of water in the vase with marbles, let's go through the problem step-by-step.
1. Determine the Radius of the Vase:
The diameter of the vase is 6 inches. The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r_{\text{vase}} = \frac{6}{2} = 3 \text{ inches}
\][/tex]
2. Calculate the Volume of the Water in the Vase:
The vase is a cylinder, and the volume [tex]\( V \)[/tex] of a cylinder is given by the formula:
[tex]\[
V_{\text{water}} = \pi r_{\text{vase}}^2 h
\][/tex]
where [tex]\( h \)[/tex] is the height of the water, which is 12 inches:
[tex]\[
V_{\text{water}} = \pi (3 \text{ in})^2 (12 \text{ in}) = \pi (9) (12) = 108\pi \text{ cubic inches}
\][/tex]
3. Determine the Radius of Each Marble:
Each marble has a diameter of 3 inches. The radius [tex]\( r \)[/tex] of a marble is:
[tex]\[
r_{\text{marble}} = \frac{3}{2} = 1.5 \text{ inches}
\][/tex]
4. Calculate the Volume of One Marble:
The volume [tex]\( V \)[/tex] of a sphere (which is the shape of the marbles) is given by:
[tex]\[
V_{\text{marble}} = \frac{4}{3} \pi r_{\text{marble}}^3
\][/tex]
[tex]\[
V_{\text{marble}} = \frac{4}{3} \pi (1.5 \text{ in})^3
\][/tex]
5. Calculate the Total Volume of All Marbles:
There are 7 marbles, so the total volume is:
[tex]\[
V_{\text{total\_marbles}} = 7 \times \frac{4}{3} \pi (1.5 \text{ in})^3
\][/tex]
6. Determine the Formula for the Actual Volume of Water in the Vase:
To get the actual volume of water, we subtract the total volume of the marbles from the volume of the water:
[tex]\[
V_{\text{actual\_water}} = V_{\text{water}} - V_{\text{total\_marbles}}
\][/tex]
Plugging in the values, we get:
[tex]\[
V_{\text{actual\_water}} = \pi (3 \text{ in})^2 (12 \text{ in}) - 7 \left( \frac{4}{3} \pi (1.5 \text{ in})^3 \right)
\][/tex]
Among the given options, the correct one is:
[tex]\[
\pi (3 \text{ in})^2 (12 \text{ in}) - 7 \left( \frac{4}{3} \pi (1.5 \text{ in})^3 \right)
\][/tex]
