A right triangle has hypotenuse of length 20 . a. Let a and...

A right triangle has hypotenuse of length 20 .
a. Let $a$ and $b$ denote the lengths of the legs. Use the fact that the hypotenuse is 20 to express $b$ in terms of $a$.
b. Express the area in terms of $a$ alone.
c. Find the lengths of the legs giving maximum area.

Key Concepts

Area Formula for a Triangle

The area of a triangle is commonly calculated as one half the product of its base and height. In the context of a right triangle, the two legs serve as the base and height. This formula is crucial when expressing the area in terms of a single variable to prepare for optimization.

Optimization Using Calculus

Optimization using calculus involves finding the maximum or minimum value of a function. This is typically done by expressing the quantity of interest (like area) in terms of one variable, differentiating with respect to that variable, and then solving for when the derivative equals zero. This method helps identify the optimal dimensions that maximize or minimize the desired quantity.

Pythagorean Theorem

This theorem establishes the relationship between the sides of a right triangle: the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. It enables one to express one leg in terms of the hypotenuse and the other leg and is a foundational principle in geometry.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate a variable of interest. In problems involving geometric shapes, such as expressing one side of a triangle in terms of another, algebraic manipulation is essential to simplify equations and make further analysis, such as optimization, feasible.

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