Question

Cups A and B are cone shaped and have heights of `33 cm` and `37 cm` and openings with radii of `10 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

`color(blue)(h~~27.028" cm to 3 decimal places")`

Explanation:

`color(blue)("Test condition")`

Consider across section vertically through the centre of each cone.

If the area of the taller cross section will fit in the area of the shorter one then so will the volumes.

Let cross section area be `A`

we require that `A_B < A_A` ( area for `A_A` is bigger )

`"require that "7xx37< 10xx33`

`259 < 330` The test condition is true

`color(red)("It will fit!")`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("To find volume of cone B")`
Volume a circular cone is `1/3xx"base area"xx"height"`

Thus the volume of B is: `" "1/3 pi (7)^2 xx37cm^3`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("To find height within cone A")`

The radius will vary in accordance with the height from zero to full radius at the top height of 33cm.

So `color(green)("radius")` at any height is :`r/33xxh = color(green)(10/33xxh)`

The volume of the transferred material is`" "1/3 pi (7)^2 xx37cm^3`

So `cancel(1/3pi )(7)^2 xx37cm^3=cancel(1/3pi) (color(green)(10/33xxh))^2xx h`

`=>49xx37=color(green)(100/1089xxh^2)xxh`

`=>h= root(3)((49xx37xx1089)/100)`

`color(blue)(h~~27.028" cm to 3 decimal places")`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Answer 2

height = `27.03cm`

Explanation:

We need to find and compare the Volumes of A and B.
Check first whether they are similar in shape - this would make some of the calculations easier.

Are the sides in the same ratio?
`7/10 =0.7 and 37/33 =1.12 " "rArr " A and B are not similar"`

`Vol_("cone") = (pi r^2 h)/3`

`Vol_A = (pi xx10^2xx33)/3" "and " "Vol_B = (pixx7^2xx37)/3`

Which is bigger? We do not need to include `pi and 3` in the comparison because they are common.

`color(white)(xxxxxxxxxx)10^2 xx33 > 7^2 xx37`

Therefore A will not overflow but we need to find the height.

The cone formed by the water in A and the whole cone of A are similar in shape.

The ratio of the cubes of the heights is equal to the ratio of the volumes.

`color(white)(xxxxxxxxxxxxxxxxx)h^3/H^3 = v/V`

`h^3/33^3 = (7^2 xx37)/(10^2 xx 33)`

`h^3 = (33^3 xx7^2 xx37)/(10^2xx33) = 19,743.57`

`h = root3(19,743.57)`

`27.03cm`