### Introducing Calculus

Many ways problem

## Solution

What is the greatest area of a rectangle inscribed inside a given right-angled triangle?

Consider this situation, where C is a vertex of both the rectangle and the triangle.

This problem can be tackled in many ways, some of which are more effective than others.

We might consider an algebraic approach.

One of the first things we must do when taking an algebraic approach is to decide which length in the diagram to consider as our variable. The suggestion in the main problem page suggested that we choose the horizontal length of the rectangle, $x$. However, we could just have easily chosen the vertical length of the rectangle. We may have even chosen to work with one of the other lengths in the diagram.

So how do we know which will be the best choice of variable? We might assume that it will make no difference to the final result but does it make a difference to the journey? Will one choice of variable be ‘easier’ to work with than another?

Alternatively we might consider a possible geometric approach.

You might like to reconsider your approach to this problem after exploring some of the ideas at Calculus of Powers.