Question

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?

Answer 1

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 `6 1/2`cm

Explanation:

Let's start with the equation for the Volume of a cone:

`V=1/3pir^2h`

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:

`V=1/3pi7^2(26)`

`V=1/3pi(49)(26)`

`V=(49*26)/3pi` - I'm going to leave it in this form for now as we work with cone A.

So let's first prove definitively that the volume of cone A is greater than cone B:

`V=1/3pi(14)^2(33)`

`V=1/3pi(196)(33)`

`V=(196*33)/3pi`

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 `(33>26)`, so cone B won't overflow cone A.

So how high up will the fluid come up? Let's solve cone A for height with the volume of cone B:

`V=1/3pir^2h`

`(49*26)/3pi=1/3pi14^2h`

`h=(49*26)/3pi*3/(pi14^2)=13/2`

So the water will rise to a height of `6 1/2`cm