A cylindrical vase has a diameter of 6 inches. At the bottom of the vase, there are 7 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? π(12in)2(3in) − 7(π(1.5in)3) π(3in)2(12in) − 7(π(1.5in)3) π(12in)2(3in) − 1.5(π(7in)3) π(3in)2(12in) − 1.5(π(7in)3)

The correct formula to calculate the volume of water in the vase is π(3in)²(12in) − 7(π(1.5in)³).

Here's the step-by-step explanation:

1. The volume of a cylinder (which is the shape of the vase) is given by the formula V = πr²h, where r is the radius and h is the height. The radius of the vase is half of the diameter, so r = 6in/2 = 3in. The height of the water is given as 12in. So, the total volume of the vase filled with water and marbles is V = π(3in)²(12in).

2. The volume of a sphere (which is the shape of a marble) is given by the formula V = 4/3πr³, where r is the radius. The radius of a marble is half of the diameter, so r = 3in/2 = 1.5in. The total volume of the 7 marbles is therefore 7 * V = 7(4/3π(1.5in)³). However, in the options given, the 4/3 factor is missing, so we assume that the volume of a marble is approximated by V = π(1.5in)³.

3. To find the volume of the water alone, we subtract the total volume of the marbles from the total volume of the vase filled with water and marbles. This gives us the formula π(3in)²(12in) − 7(π(1.5in)³).