Math CentralQuandaries & Queries


Question from Elizabeth, a student:

If i have a cup that holds a volume of 477mL and the bottom radius is 2.8cm and the top radius is 4.9cm and the height is 10cm. If i increase the volume by one and a half times what is the new measurements if the cup is directly proportional to the first one.
Thank you

Hi Elizabeth,

Suppose the cup has top radius R cm bottom radius r cm and height h cm. Write an expression for the volume in terms of R, r and h. If the dimensions of the new cup are directly proportional to the dimensions of the old cup then there is a number k so that the radius of the top of the new cup is kR cm, the radius of the bottom is kr cm and the height is kh cm. Write an expression for the volume of the new cup in terms of kR, kr and kh. Use the fact that

volume of the new cup = 1.5 (volume of the old cup)

and solve for k.

I hope this helps,


Hi Elizabeth.

My instinctive response to "scale" questions is this: as linear measurements scale linearly, areas scale quadraticly and volumes scale cubicly.

You multiplied the volume by 2.5. (When you "increase by one and a half times" you have the original 1 + the increase of 1.5 = 2.5 times the original size. If you meant to say you increase it by half, then you'd be multiplying the volume by 1.5 instead of 2.5.)

That means the cube root of 2.5 is the factor by which you multiply the linear dimensions. The cube root of 2.5 is about 1.3572.

Thus the big cup would have a bottom radius of 2.8cm(1.3572), a top radius of (4.9)(1.3572) and a height of 10(1.3572).

Stephen La Rocque.

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