Problem 6 - Calculus for Biology and Medicine

Question

Suppose that \(a\) and \(b\) are the side lengths in a right triangle whose hypotenuse is \(5 \mathrm{~cm}\) long. What is the largest perimeter possible?

Step-by-Step Solution

Answer

The largest possible perimeter is \(5\sqrt{2} + 5\) cm when \(a = b = \frac{5\sqrt{2}}{2}\).

Step 1: Understanding the Problem

We are given a right triangle with side lengths \(a\) and \(b\), and a hypotenuse of \(5\) cm. Our task is to find the largest possible perimeter of this triangle.

Step 2: Using the Pythagorean Theorem

According to the Pythagorean theorem, in a right triangle, the relationship between the sides is given by \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse. Since the hypotenuse is \(5\) cm, we have \(a^2 + b^2 = 5^2 = 25\).

Step 3: Express Perimeter Formula

The perimeter \(P\) of the triangle is the sum of all sides: \(P = a + b + c\). Since \(c = 5\), the perimeter becomes \(P = a + b + 5\).

Step 4: Optimize \(a + b\) Subject to Pythagorean Constraint

We want to maximize \(a + b\) given the equation \(a^2 + b^2 = 25\). To maximize \(a + b\), use the substitution \(x = a + b\) and \(y = ab\), noting the identity \((a + b)^2 = a^2 + 2ab + b^2\). Substituting the known equality gives \(x^2 - 2y = 25\).

Step 5: Application of AM-GM Inequality

Using the AM-GM inequality on \(a^2 + b^2 \geq 2ab\) gives \(25 \geq 2ab\), or \(ab \leq \frac{25}{2}\). To maximize \(a + b\), try when \(a = b\).

Step 6: Equal Side Lengths and Solve for \(a\) and \(b\)

If \(a = b\), substituting in the Pythagorean theorem results in \(2a^2 = 25\). Solving this, we get \(a = b = \frac{5}{\sqrt{2}}\) or \(a = b = \frac{5\sqrt{2}}{2}\).

Step 7: Calculate Maximum Perimeter

Substitute \(a = b = \frac{5\sqrt{2}}{2}\) into the formula for the perimeter: \(P = a + b + 5 = \frac{5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2} + 5 = 5\sqrt{2} + 5\).

Step 8: Conclusion

Therefore, the largest possible perimeter of the triangle, considering equal side lengths for maximum sum \(a+b\), is \(5\sqrt{2} + 5\).

Key Concepts

Pythagorean Theorem

In a right triangle, the Pythagorean Theorem is a fundamental tool that relates the lengths of the sides. This theorem states: \( a^2 + b^2 = c^2 \), where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse. This relationship allows us to solve for any unknown side if the other two are known.
For our problem, the hypotenuse \(c\) is given as 5 cm. Applying the Pythagorean Theorem, we derive that \(a^2 + b^2 = 25\). This equation lays the groundwork for further calculations, such as optimizing the perimeter of the triangle.
Understanding this theorem is crucial because it provides a direct means to relate and calculate side lengths in right triangles. This connection is essential for many mathematical and real-world applications.

Perimeter Calculation

The perimeter of a triangle is the total distance around the triangle, calculated by adding the lengths of all its sides. For a right triangle with sides \(a\), \(b\) and hypotenuse \(c\), the perimeter \(P\) is expressed as: \(P = a + b + c\).
In our case, since \(c = 5\), the perimeter is simplified to \(P = a + b + 5\).
Maximizing the perimeter involves making \(a + b\) as large as possible within the constraints given by the Pythagorean Theorem, \(a^2 + b^2 = 25\). This approach leverages the structure of the triangle and ensures efficient calculation without unnecessary complexity.
Perimeter calculations have broader utility in many areas, including geometry, construction, and design, where understanding spatial dimensions is crucial.

AM-GM Inequality

The AM-GM Inequality is a powerful mathematical concept used to relate the arithmetic mean and geometric mean of non-negative numbers. It states that for any non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. For two numbers, \( a \) and \( b \), the inequality is: \( \frac{a + b}{2} \geq \sqrt{ab} \).
In the context of our exercise, we use the AM-GM inequality to help maximize \(a + b\) under the constraint \(a^2 + b^2 = 25\). Applying the inequality gives: \( a^2 + b^2 \geq 2ab \), thus \( 25 \geq 2ab \) or \( ab \leq 12.5 \).
This insight is pivotal because it aids in checking the feasibility of potential side lengths \(a\) and \(b\) by ensuring they adhere to this mathematical bound while working to maximize their sum.
Such inequalities are vital in optimization problems across various fields, enabling solutions that maximize or minimize particular quantities efficiently.