Math CentralQuandaries & Queries


Question from Tehmas, a student:

The dimensions of a gift box are consecutive positive integers such that the height is the lowest integer and the length is the greatest integer. If the height is increased by 1cm, the width increased by 2 cm, and the length increased by 3 cm, then a larger box is constructed and the volume is increased by 456 cm^3. Determine the dimensions of each box.

Hi Tehmas,

Suppose the height of the gift box is $x$ cm then, since the dimensions are consecutive positive integers, the width is $x + 1$ cm and the length is $x + 2$ cm. Thus the volume of the gift box is $x(x + 1)(x + 2) \mbox{ cm}^3.$

What are the dimensions of the larger box? What is the expression for its volume?

The problem states that

\[ \mbox{Volume of the larger box } = x(x + 1)(x + 2) + 456 \mbox{ cm}^3\]

Solve for $x.$

When I did this the value of $x$ that I obtained was not an integer. Are you sure that you have the problem stated correctly?


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