# Thread: need help with an Optimization

1. ## need help with an Optimization Problem

This question confuse me a bit out of all the questions im suppose to do, and hopefully someone can help me.

"A rectangle is inscribed in a right triangle with sides 6,8, and 10. Two sides of the rectangle lie along legs of teh triangle, and teh remaining vertex of the rectangle is on the hypotenuse of the triangle. What dimensions for the rectangle will maximize its area?"

2. Originally Posted by katoru007
This question confuse me a bit out of all the questions im suppose to do, and hopefully someone can help me.

"A rectangle is inscribed in a right triangle with sides 6,8, and 10. Two sides of the rectangle lie along legs of teh triangle, and teh remaining vertex of the rectangle is on the hypotenuse of the triangle. What dimensions for the rectangle will maximize its area?"
Units??

Let the side that lies on the leg of length 8 be of length x.

Let the side that lies on the leg of length 6 be of length y.

Then A = xy.

Now note that there are two smaller right-triangles on two sides of the rectangle. These triangles are similar to the 6-8-10 right-triangle.

So you have a choice of restriction:

$\displaystyle \frac{y}{8-x} = \frac{6}{8} \Rightarrow y = \, ....$

or

$\displaystyle \frac{6-y}{x} = \frac{6}{8} \Rightarrow x = \, ....$

Use one of these restrictions to write A in terms of a single variable. Now apply calculus to find the answer.

3. thank you very much. =]