Cups A and B are cone shaped and have heights of #33 cm# and #26 cm# and openings with radii of #14 cm# and #7 cm#, respectively. If cup B is full and its contents are poured into cup A, will cup A overflow? If not how high will cup A be filled?
1 Answer
Solve for the volume of B and use that to help solve for the height of A to get that the water will rise to a height of
Explanation:
Let's start with the equation for the Volume of a cone:
We're being asked to determine if the volume of cone B is greater than cone A (will it overflow from the contents of cone B). Just looking at the measurements of the 2 cones, with the height of cone A and it's radius being bigger than cone B, it's pretty clear the volume of cone B is smaller than cone A. So the next part of the question asks how high the water will come up in cone A.
So let's first determine the volume of cone B:
So let's first prove definitively that the volume of cone A is greater than cone B:
Again, we can see that cone A has the greater volume: the bigger term of A (196) is greater than the bigger term of B (49), as is the smaller term
So how high up will the fluid come up? Let's solve cone A for height with the volume of cone B:
So the water will rise to a height of