A cylindrical vase has a diameter of 6 inches. At the bottom of the vase there are 9 marbles, each of diameter 3 inches. The vase is filled with water up to a height of 12 inches. Which of the following could be used to calculate the volume of water in the vase? \( \pi(3 \mathrm{in})^{2}(12 \mathrm{in})-1.5\left(\frac{4}{3} \pi(9 \mathrm{in})^{3}\right) \) \( \pi(12 \mathrm{in})^{2}(3 \mathrm{in})-1.5\left(\frac{4}{3} \pi(9 \mathrm{in})^{3}\right) \) \( \pi(3 \mathrm{in})^{2}(12 \mathrm{in})-9\left(\frac{4}{3} \pi(1.5 \mathrm{in})^{3}\right) \) \( \pi(12 \mathrm{in})^{2}(3 \mathrm{in})-9\left(\frac{4}{3} \pi(1.5 \mathrm{in})^{3}\right) \)

To solve this delightful puzzle, you first need to calculate the volume of water in the vase. The formula for the volume of a cylinder is \( \pi r^2 h \). With a diameter of 6 inches, the radius is 3 inches, and the height of the water is 12 inches. So, you calculate the volume of water as \( \pi(3 \, \text{in})^2(12 \, \text{in}) \). Next, consider the volume displaced by the marbles. Since each marble has a diameter of 3 inches, the radius is 1.5 inches. The volume of one marble is given by \( \frac{4}{3} \pi (1.5 \, \text{in})^3 \), and with 9 marbles, you multiply this volume by 9. To find the actual volume of water, you need to subtract the total volume occupied by the marbles from the initial volume of water in the vase. So, the correct option involves calculating these volumes while properly accounting for the displacement by the marbles! Fun yet mind-bending, right? If you're curious about the math behind the scenes, dive into resources on volume geometry; they're abundant with engaging visuals and examples! For a hands-on twist, consider conducting simple experiments with vases and marbles to visualize these principles—you might just become a volume wizard!