Question

The product of the base and height of a rectangle is the area. Can this statement be represented as a direct variation for the base and height? (Hint: Treat the area as a constant.)

Answer 1

A direct variation is given by the following formula:

$x \times k = y$

Direct variation involves multiplying a variable by a constant, k, to equal a variable. This means that both variables increase with each other.

The area of a rectangle is given by the following formula:

$A = bh$

This is an inverse variation, because if A was a constant, b and h would have to multiply each other to equal the constant, which means that b and h increase and decrease away from each other. This means that you cannot represent this statement as a direct variation.

Answer 2

Look at the formula A = b(h):  Area of a rect. = base times height.

In the unusual situation where you'd hold the area constant and vary b and h, then   A=bh could be re-written as

 A
---- = h    or  A/h = b.  In both cases we'd be working with inverse proportion,
 b                                   not direct proportion / variation.