Question

A glassmaker is making a cuboidal glass container of volume 1080 cubic inches. If the container has a square base, how many such containers can he make? (All edges are measured in natural numbers in inches.)

• A. 4
• B. 6
• C. 8
• D. 2

Answer 1

The correct option is A 4

The base of the container must be a square. So, we can take the area of the base as a perfect square.

$Areaofthesquarebase=S^2$

Now, the volume of the container will be:

$Volume=Areaofthebase\times Height⟹1080cuin=S^2\times Height$

So, the number of different containers will be the number of ways 1080 can be expressed as the product of a perfect square and another natural number.

In other words, the number of different containers will be same as the number of factors of 1080, which are also perfect squares.

Now, factorising 1080, we find:

$1080=2^3\times 3^3\times 5$

For a perfect square, the powers of each prime factor have to be an even number.

Hence, the perfect square factors of 1080 are:

$\left(i\right)2^0\times 3^0\times 5^0=1\left(ii\right)2^2\times 3^0\times 5^0=4\left(iii\right)2^0\times 3^2\times 5^0=9\left(iv\right)2^2\times 3^2\times 5^0=36$

Therefore, the glassmaker will be able to make 4 distinct containers, all with a square base and a volume of 1080 cu in.