Express the area of the shaded region as a function of x.

[Zavelin]

Having a hard time working out this problem. I've tried working through by manipulating the area and perimeter equations for rectangles (A = lw, P = 2w + 2l), but I'm just not getting that "AHA!" moment!

A push in the right direction would be appreciated.


Problem: Express the area A of the shaded region shown to the right as a function of x and y.





Thanks in advance for any hints and tips anyone can offer.

 

[soroban]

Hello, Zavelin!

Express the area of the shaded region as a function of \(\displaystyle x\) and \(\displaystyle y.\)

      * - - - * - - - *
      |       |:::::::|
      |      x|:::::::| 
      |   y   |:::::::|
      * - - - *:::::::|
      |:::::::::::::::|
   20 |:::::::::::::::|
      |:::::::::::::::|
      * - - - - - - - *
             40

The width of the entire rectangle is \(\displaystyle 40\); its height is \(\displaystyle x+20\).
. . Its area is: .\(\displaystyle 40(x+20)\)

The width of the upper-left corner is \(\displaystyle y\); its height is \(\displaystyle x\).
. . Its area is: .\(\displaystyle xy\).

The shaded area is: .\(\displaystyle 40(x+20) - xy \:=\:40x + 800 - xy\)

 

[Zavelin]

Thank. You. So. Much!

And here I was splitting the darn thing into triangles, trying to apply Pythagorean Theorem. HA!