Question

a cylindrical vase has a diameter of 4 inches. at the bottom of the vase, there are 6 marbles, each of diameter 3 inches. the vase is filled with water up to a height of 8 inches. which of the following could be used to calculate the volume of water in the vase?
π(8in)²(2in) - 1.5(4/3π(6in)³)
π(2in)²(8in) - 1.5(4/3π(6in)³)
π(8in)²(2in) - 6(4/3π(1.5in)³)
π(2in)²(8in) - 6(4/3π(1.5in)³)

Explanation:

Step1: Calculate cylinder volume

The volume formula for a cylinder is $V = \pi r^{2}h$. The diameter of the vase is 4 inches, so the radius $r = 2$ inches and the height of water $h = 8$ inches. The volume of the cylindrical - part filled with water is $V_{cylinder}=\pi(2\text{in})^{2}(8\text{in})$.

Step2: Calculate marble volume

The volume formula for a sphere is $V_s=\frac{4}{3}\pi r^{3}$. The diameter of each marble is 3 inches, so the radius of each marble $r = 1.5$ inches. There are 6 marbles, so the total volume of the 6 marbles is $V_{marbles}=6\times\frac{4}{3}\pi(1.5\text{in})^{3}$.

Step3: Calculate water volume

The volume of water $V_{water}=V_{cylinder}-V_{marbles}=\pi(2\text{in})^{2}(8\text{in})-6\times\frac{4}{3}\pi(1.5\text{in})^{3}$.

Answer:

$\pi(2\text{in})^{2}(8\text{in})-6\times\frac{4}{3}\pi(1.5\text{in})^{3}$