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.

Question

Which of the following could be used to calculate the volume of water in the vase? (4 points)

π(12in)2(3in) − 7(four over threeπ(1.5in)3)

π(3in)2(12in) − 7(four over threeπ(1.5in)3)

π(12in)2(3in) − 1.5(four over threeπ(7in)3)

π(3in)2(12in) − 1.5(four over threeπ(7in)3)

Answer 1

To calculate the volume of water in the cylindrical vase, we need to determine the volume of the vase filled with water and subtract the volume displaced by the marbles at the bottom.

1. Volume of the cylindrical vase: The formula for the volume \( V \) of a cylinder is given by: \[ V = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.

   The diameter of the vase is 6 inches, so the radius \( r \) is: \[ r = \frac{6 \text{ in}}{2} = 3 \text{ in} \] The height \( h \) of the water is 12 inches. Therefore, the volume of water in the vase is: \[ V_{\text{water}} = \pi (3 \text{ in})^2 (12 \text{ in}) = \pi(3^2)(12) = \pi(9)(12) = 108\pi \text{ in}^3 \]

2. Volume of the marbles: Each marble is in the shape of a sphere, and the volume \( V \) of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] The diameter of each marble is 3 inches, thus the radius \( r \) is: \[ r = \frac{3 \text{ in}}{2} = 1.5 \text{ in} \] Therefore, the volume of one marble is: \[ V_{\text{marble}} = \frac{4}{3} \pi (1.5 \text{ in})^3 \] The total volume of 7 marbles is: \[ V_{\text{marbles}} = 7 \left(\frac{4}{3} \pi (1.5)^3\right) \]

3. Total volume of water in the vase: To find the volume of water alone, we subtract the volume of the marbles from the volume of the water in the vase: \[ V_{\text{water}} = V_{\text{cylinder}} - V_{\text{marbles}} \] Therefore, we get: \[ V_{\text{water}} = \pi (3 \text{ in})^2 (12 \text{ in}) - 7 \left(\frac{4}{3} \pi (1.5 \text{ in})^3\right) \]

Now comparing the options provided:

1. \( \pi(12 \text{ in})^2(3 \text{ in}) - 7\left(\frac{4}{3}\pi(1.5 \text{ in})^3\right) \)
2. \( \pi(3 \text{ in})^2(12 \text{ in}) - 7\left(\frac{4}{3}\pi(1.5 \text{ in})^3\right) \) (This is the correct expression)
3. \( \pi(12 \text{ in})^2(3 \text{ in}) - 1.5\left(\frac{4}{3}\pi(7 \text{ in})^3\right) \)
4. \( \pi(3 \text{ in})^2(12 \text{ in}) - 1.5\left(\frac{4}{3}\pi(7 \text{ in})^3\right) \)

Thus, the correct formula to calculate the volume of water in the vase is: \[ \pi(3 \text{ in})^2(12 \text{ in}) - 7\left(\frac{4}{3}\pi(1.5 \text{ in})^3\right) \]

Therefore, the correct option is: π(3in)²(12in) − 7\left(\frac{4}{3}\pi(1.5in)³\right).