Question

Maximizing Area What is the largest possible area for a right triangle whose hypotenuse is 5 cm long, and what are its dimensions?

Short answer

The largest possible area for a right triangle with a hypotenuse of length 5 cm is approximately \( 6.25 \) square cm and the dimensions are approximately \( 3.54 \) cm and \( 3.54 \) cm.

Step-by-step solution

Define the sides of the triangle

Let one of the sides of the triangle be \( x \). Then, according to the Pythagorean theorem, the other side will be \( \sqrt{5^2 - x^2} \) or \( \sqrt{25 - x^2} \).

Formulate the Area function

Next, we form the Area function of the triangle, which is given by 0.5 * base * height. In this case, the base is \( x \) and the height is \( \sqrt{25 - x^2} \). Therefore, the Area \( A(x) = 0.5 * x * \sqrt{25 - x^2} \).

Maximize the Area function

To maximize the area, differentiate the function \( A(x) \) with respect to \( x \) and set to zero, then solve for \( x \). This involves applications of the product and chain rule from calculus.

Determine dimensions for maximum area

Once you have determined the \( x \) that maximizes the area, substitute this value back into the expressions for the sides of the triangle to find their lengths.

Key concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, particularly in the study of right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as:\[ c^2 = a^2 + b^2 \] where \( c \) is the length of the hypotenuse and \( a \) and \( b \) are the lengths of the other two sides. This theorem is pivotal when solving problems related to right triangles, including finding missing side lengths and understanding the relationship between the sides.
Utilizing this theorem, one can deduce the necessary dimensions of a right triangle when the length of the hypotenuse is known. In the context of maximizing areas, knowing the sides of the triangle allows us to construct an area function based on these dimensions, leading to an optimization problem solvable with the tools of calculus.

Area Function

An area function expresses the area of a shape as a function of one or more of its dimensions. For a right triangle, the area can be calculated as one half of the product of its base and height. The formula is:\[ A = \frac{1}{2} \times \text{base} \times \text{height} \] In the case of the triangle in our exercise, we define the area function using the known side length \( x \) and the expression for the other side derived from the Pythagorean theorem as \( \text{height} = \( \)sqrt{25 - x^2} \). The resulting area function for our problem becomes:\[ A(x) = \frac{1}{2} \times x \times \( \)sqrt{25 - x^2} \] This function encapsulates how the area of the triangle changes as one of the sides varies, holding the hypotenuse constant, and is essential for identifying the dimensions that yield the maximum area.

Differentiation

Differentiation is a core operation in calculus that calculates the rate at which a function changes at any point. It provides a way to determine the slope of the tangent to the curve of a function at a given point. When we differentiate an area function with respect to one of its sides, we are seeking to understand how a small change in that side length will affect the change in area.
In optimization problems, differentiation is used to find the maximum or minimum values of a function. After finding the derivative of the area function, a critical step is to set the derivative equal to zero and solve for the variable in question. The values obtained represent points where the function has a horizontal tangent line, which could be indicative of maximum or minimum points. This step is integral to the solution in finding the largest possible area for the right triangle.

Product Rule

The product rule is a derivative rule used when differentiating functions that are products of two or more functions. In calculus, when given a function that is the product of two sub-functions, the product rule states:\[ (f(x)g(x))' = f'(x)g(x) + f(x)g'(x) \] where \( f(x) \) and \( g(x) \) are differentiable functions and \( f'(x) \) and \( g'(x) \) are their respective derivatives. This rule allows one to calculate the derivative of complex functions by breaking them down into simpler parts.
In our example, when we maximize the area function of the right triangle, the product rule is applied to differentiate the function \( A(x) = 0.5 \times x \times \( \)sqrt{25 - x^2} \). This step is fundamental to discover the point at which the area reaches its maximum value.

Chain Rule

The chain rule is another essential differentiation technique used when a function consists of another function nested within it — a composition of functions. It can be thought of as finding the derivative 'outside in.' The rule is formally expressed as:\[ (f(g(x)))' = f'(g(x)) \times g'(x) \] where \( f \) is a function and \( g \) is another function inside \( f \).
For instance, in our area maximization problem, the function \( \( \)sqrt{25 - x^2} \) is nested inside the area function, making it a composite function that requires the chain rule for proper differentiation. The chain rule allows us to unravel the derivative of this composite function, leading us towards identifying the maximum area of the triangle.