# Thread: Optimizing Problem, max/min, rectangle inside a triangle

1. ## Optimizing Problem, max/min, rectangle inside a triangle

Find the area of the largest rectangle that can be inscribed in a right triangle with legs adjacent to the right angle of lengths 5cm and 12cm. The two sides of the rectangle lie along the legs.

2. Originally Posted by ellenberger
Find the area of the largest rectangle that can be inscribed in a right triangle with legs adjacent to the right angle of lengths 5cm and 12cm. The two sides of the rectangle lie along the legs.
Ellen:

How much progress have you made with this problem?

Let x and y be the sides.

x will be the side of the rectangle along the triangle side of length 5, and y will be the side along the triangle side of length 12.

Identify the right triangle (lower right if side of length 5 is the base) with legs (5-x) and y. This triangle is similar to the big right triangle, so the following ratio is valid:

$\displaystyle \frac{y}{12} = \frac{5-x}{5}$

Solve for y:

$\displaystyle y = 12 - \frac{12x}{5}$ (*)

The objective function is the area of the rectangle:

$\displaystyle A = x y$

Write this area as a single variable function by using substitution (*) above, then apply differential calculus techniques to optimize.

I hope this helps.

Good luck!

3. Originally Posted by apcalculus
Ellen:

How much progress have you made with this problem?

Let x and y be the sides.

x will be the side of the rectangle along the triangle side of length 5, and y will be the side along the triangle side of length 12.

Identify the right triangle (lower right if side of length 5 is the base) with legs (5-x) and y. This triangle is similar to the big right triangle, so the following ratio is valid:

$\displaystyle \frac{y}{12} = \frac{5-x}{5}$

Solve for y:

$\displaystyle y = 12 - \frac{12x}{5}$ (*)

The objective function is the area of the rectangle:

$\displaystyle A = x y$

Write this area as a single variable function by using substitution (*) above, then apply differential calculus techniques to optimize.

I hope this helps.

Good luck!

very helpful, thank you

4. Originally Posted by ellenberger
very helpful, thank you

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### maximum area of two adjacent rectangles in an acute angle traiangle

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