Little optimization problem

Jun 2009
58
0
Hello friends¡¡

I need to find the greatest area rectangle which can be drawn "inside" of a right triangle.

Measures: Cathetus a = 4cm and cathetus b = 3cm.

I need to solve the problem by using derivatives.

Thanks in advance
 

Soroban

MHF Hall of Honor
May 2006
12,028
6,341
Lexington, MA (USA)
Hello, osodud!

Find the greatest area rectangle which can be drawn
inside a right triangle with legs 3 and 4.
A diagram will always help.
Code:
-   - *
:   : |  *
:  3-y|     *
:   : |        *
3   - *-----------*
:   : |     x     |  *
:   y |           |y    *
:   : |           |        *
-   - *-----------*-----------*
      : - - x - - :  - 4-x -  :
      : - - - - - 4 - - - - - :

The area of the rectangle is: .\(\displaystyle A \;=\;xy\) .[1]


From the similar right triangles, we have: .\(\displaystyle \frac{3-y}{x} \:=\:\frac{y}{4-x}\)
. . which simplifies to: .\(\displaystyle y \;=\;3-\tfrac{3}{4}x\) .[2]


Substitute [2] into [1]: .\(\displaystyle A \;=\;x\left(3-\tfrac{3}{4}x\right)\)


And you must maximize: .\(\displaystyle A \;=\;3x - \tfrac{3}{4}x^2\)

. . Got it?