Question

A cylindrical vase has a diameter of 6 inches. At the bottom of the vase, there are 9 marbles, each of diameter 1 inch. The vase is filled with water up to a height of 12 inches.

What is the volume of water in the vase?
27π in3
65π in3
96π in3
105π in3

Answer 1

First, we need to find the volume of the water if there were no marbles in the vase, but the water height were the same. Once we have that volume, we need to subtract the volume of the marbles.

The formula for the volume of a cylinder is \[V=\pi \cdot h\cdot r^2\]So the volume of the ater if there were no marbles in the vase, but the water height were the same is given by the equation \[V=\pi\cdot 12\cdot 3^2 =108\pi\]

Answer 2

Now we need to find the volume of each of the marbles and subtract that off. The volume for a sphere is\[V={4\over3}\cdot \pi\cdot r^3\]We're given the diameter of the marbles to be 1, so the radius is \(1/2\). Hence, the volume of each is\[V={4 \over3}\cdot \pi\cdot \left({1\over2}\right)^3={\pi\over6}\]

Answer 3

For 9 marbles, the volume would then be \[{9\pi \over 6}={3\pi \over 2}\]Now we just need to subtract the two volumes. \[{108\pi}-{3\pi \over 2}={106.5\pi}\]

Answer 4

However, this is not one of your options. Perhaps your teacher meant to say that the marbles have a radius of 1, and not a diameter of 1?

Answer 5

no im sure because it is part of an online assignment, and i also tried just didn't reach with one of the answers.

Answer 6

The only thing that might have happened was if whoever wrote this question meant radius instead of diameter there. If that were the case, the answer would be \(96\pi\). However, as written, the solution is \[106.5\pi\]

Answer 7

well i figure it out, still thanks for your help i liked the way you explained it and took me step by step. (:

Answer 8

You're welcome :)