Let's thoroughly understand the problem and the suitable formula to solve it.
1. Volume of the Vase:
- The vase is cylindrical with a diameter of 6 inches.
- The radius of the vase, therefore, is [tex]\( \frac{6 \text{ inches}}{2} = 3 \text{ inches} \)[/tex].
- The height of the water in the vase is 12 inches.
- The formula to calculate the volume of a cylinder is given by [tex]\( V_{\text{cylinder}} = \pi r^2 h \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
- So, the volume of the water-filled part of the vase is:
[tex]\[
V_{\text{vase}} = \pi (3 \text{ inches})^2 (12 \text{ inches}) = 339.292\ \text{cubic inches}
\][/tex]
2. Volume of a Single Marble:
- Each marble has a diameter of 3 inches.
- So, the radius of each marble is [tex]\( \frac{3 \text{ inches}}{2} = 1.5 \text{ inches} \)[/tex].
- The formula for the volume of a sphere is [tex]\( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \)[/tex], where [tex]\( r \)[/tex] is the radius.
- The volume of one marble is:
[tex]\[
V_{\text{marble}} = \frac{4}{3} \pi (1.5 \text{ inches})^3 = 14.137\ \text{cubic inches}
\][/tex]
3. Total Volume of All Marbles:
- There are 9 marbles at the bottom of the vase.
- Therefore, the total volume occupied by all the marbles is:
[tex]\[
V_{\text{total marbles}} = 9 \times 14.137\ \text{cubic inches} = 127.234\ \text{cubic inches}
\][/tex]
4. Volume of Water in the Vase:
- The water occupies the space in the vase not taken up by the marbles.
- Hence, the volume of water in the vase is the volume of the vase minus the total volume of the marbles:
[tex]\[
V_{\text{water}} = V_{\text{vase}} - V_{\text{total marbles}} = 339.292\ \text{cubic inches} - 127.234\ \text{cubic inches} = 212.058\ \text{cubic inches}
\][/tex]
With the calculations verified, let's inspect the given options to identify the correct one based on the explanations above:
[tex]\[
\pi(3 \text{ in})^2(12 \text{ in}) - 9\left(\frac{4}{3} \pi(1.5 \text{ in})^3\right)
\][/tex]
The correct option is:
[tex]\[
\pi(3 \text{ in})^2(12 \text{ in}) - 9\left(\frac{4}{3} \pi(1.5 \text{ in})^3\right)
\][/tex]
Using this formula accurately calculates the volume of water in the vase, taking into account the marbles' volumes.
