The region is composed of two simple shapes A triangle with

The region is composed of two simple shapes: A triangle with a base of 2ft and a height of 2 ft and attached to the base of the triangle a rectangle 2 ft by 1 ft. Find a formula for the area shown included up to x. Note that x can vary from the bottom of the triangle up to the top of the rectangle.

Solution

Clearly first we check if x is only upto the triangle...

By similarity, we can state that...
x/2 = b/2
So, x = b here

So, the area of the triangle with ht = base = x is :
1/2*x*x = x^2/2

So, we have
A = x^2/2 if 0 < x <= 2
This is so because the triangle exists upto a height of 2

Now, if x > 2 :
Some part of the rectangle would also get involved ....

The area of the rectangle is :
(x - 2)*2
= 2x - 4

So, for 2 < x < 3, we have :
A = x^2/2 + 2x - 4

So, the final formula is :

A = x^2/2 ; 0 < x <= 2
A = x^2/2 + 2x - 4 ; 2 <= x < 3