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

• Mar 24th 2010, 12:22 PM
ellenberger
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.
• Mar 24th 2010, 04:15 PM
apcalculus
Quote:

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:

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

Solve for y:

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

The objective function is the area of the rectangle:

$A = x y$

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!
• Mar 24th 2010, 06:26 PM
ellenberger
Quote:

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:

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

Solve for y:

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

The objective function is the area of the rectangle:

$A = x y$

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!