Question

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? (pi(3\text{in})^2(12\text{in})-7(\frac{4}{3}pi(1.5\text{in})^3)) (pi(12\text{in})^2(3\text{in})-1.5(\frac{4}{3}pi(7\text{in})^3)) (pi(3\text{in})^2(12\text{in})-7(\frac{4}{3}pi(1.5\text{in})^3)) (pi(12\text{in})^2(3\text{in})-7(\frac{4}{3}pi(1.5\text{in})^3))

Explanation:

Step1: Calculate cylinder volume

The volume formula for a cylinder is $V = \pi r^{2}h$. Given the diameter of the vase is 6 inches, the radius $r=\frac{6}{2}=3$ inches and the height of water $h = 12$ inches. So the volume of the part - of the cylinder filled with water is $V_{cylinder}=\pi(3)^{2}(12)$.

Step2: Calculate marble volume

The volume formula for a sphere is $V_s=\frac{4}{3}\pi r^{3}$. Each marble has a diameter of 3 inches, so the radius of each marble $r = 1.5$ inches. The volume of one marble is $V_{one - marble}=\frac{4}{3}\pi(1.5)^{3}$, and the volume of 7 marbles is $V_{7 - marbles}=7\times\frac{4}{3}\pi(1.5)^{3}$.

Step3: Calculate water volume

The volume of water in the vase is the volume of the cylinder - shaped part filled with water minus the volume of the 7 marbles. So $V = \pi(3)^{2}(12)-7\times\frac{4}{3}\pi(1.5)^{3}$.

Answer:

$\pi(3\text{in})^{2}(12\text{in})-7(\frac{4}{3}\pi(1.5\text{in})^{3})$