Question

question 6(multiple choice worth 4 points)
(06.02 lc)
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?
o π(3in)^2(12in) - 4/3 π(1.5in)^3
o π(3in)^2(12in) - 9(4/3 π(1.5in)^3)
o π(12in)^2(3in) - 1.5(4/3 π(3in)^3)
o π(12in)^2(3in) - 9(4/3 π(3in)^3)

Explanation:

Step1: Calculate the volume of the cylinder

The formula for the volume of a cylinder is \(V = \pi r^{2}h\). Given the diameter of the vase \(d = 6\) inches, so the radius \(r=\frac{d}{2}=3\) inches and the height \(h = 12\) inches. Then the volume of the cylinder \(V_{cylinder}=\pi\times(3)^{2}\times12=\pi(3^{2})(12)\).

Step2: Calculate the volume of one marble

The formula for the volume of a sphere is \(V=\frac{4}{3}\pi r^{3}\). Given the diameter of a marble \(d = 3\) inches, so the radius \(r = 1.5\) inches. Then the volume of one marble \(V_{marble}=\frac{4}{3}\pi(1.5)^{3}\).

Step3: Calculate the volume of 9 marbles

Since there are 9 marbles, the total volume of marbles \(V_{9 - marbles}=9\times\frac{4}{3}\pi(1.5)^{3}=12\pi(1.5)^{3}\).

Step4: Calculate the volume of water

The volume of water \(V_{water}=V_{cylinder}-V_{9 - marbles}\), substituting the values we get \(V_{water}=\pi(3^{2})(12)-12\pi(1.5)^{3}=\pi(3^{2})(12)-\frac{4}{3}\pi(9)(1.5)^{3}=\pi(3^{2})(12)-\frac{4}{3}\pi( ( \frac{3}{2})^{3}\times9)=\pi(3^{2})(12)-\frac{4}{3}\pi(9)(\frac{3^{3}}{2^{3}})\) (another way to look at it: \(V_{water}=\pi(3^{2})(12)-9\times\frac{4}{3}\pi(1.5)^{3}=\pi(3^{2})(12)-12\pi(1.5)^{3}=\pi(3^{2})(12)-\frac{4}{3}\pi(9)(1.5)^{3}\))

Answer:

\(\pi(3^{2})(12)-\frac{4}{3}\pi(9)(1.5)^{3}\) (which is equivalent to \(\pi(3in)^{2}(12in)-\frac{4}{3}\pi(9)(1.5in)^{3}\))