Question

Cups A and B are cone shaped and have heights of `22 cm` and `23 cm` and openings with radii of `6 cm` and `12 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

`color(brown)("Cone B has both a larger base diameter and greater height.")`
`color(brown)("So it is evident that it must have the larger volume. In")` `color(brown)("conclusion, cup A will overflow.")`

Explanation:

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Difference in volume demonstrated")`

Volume of a cone is `1/3" base area "xx" height"`

Cone A base area`" " pir^2" "->" "pi6^2" "->" "36pi`
Cone A height 22cm
`=>"Volume cone A ="36pixx22 = 792pi cm^3`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Cone B base area`" " pir^2" "->" "pi12^2" "->" "144pi`
Cone B height 23cm
`=>"Volume cone B ="144pixx23=3312pi cm^3`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Clearly cup B has a significantly larger volume.

So as the question posed stands cup A will overflow.

`color(green)("However, my experience of questions leads me to ask; should it be")` `color(green)("cup A that is bigger? Otherwise there is no point in even")` `color(green)("mentioning about height of fill level.")`