Question

A wooden beam has a rectangular cross section of height \(h\) and width \(w\) (see figure). The strength \(S\) of the beam is directly proportional to its width and the square of its height. What are the dimensions of the strongest beam that can be cut from a round log of diameter 24 inches? (Hint: \(S=k h^{2} w\), where \(k>0\) is the proportionality constant.)

Short answer

The dimensions of the strongest beam that can be cut from a round log of diameter 24 inches are height of approximately 3.46 inches and width of approximately 6.93 inches.

Step-by-step solution

Formulating the Constraint

Since the rectangular cross section is inscribed within the circular log, by using the Pythagorean Theorem, we can formulate a constraint relation between the height and width of the rectangular beam. If we consider the rectangle's diagonal as the diameter of the circle, then, \(h^{2} + w^{2} = (2 * 12)^{2} = 576\) (as radius of log is 12 inches).

Express Width in Terms of Height

From the constraint relation derived above, we can express width \(w\) in terms of height \(h\). It results in: \(w = \sqrt{576 - h^{2}}\).

Formulating Strength Function

The strength function \(S = k h^{2} w\). Substituting \(w = \sqrt{576 - h^{2}}\) into the strength function, we get \(S = k h^{2} \sqrt{576 - h^{2}}\). This is simplified to \(S = 24 k h^{2} - k h^{4}\). This is the function we want to optimize.

Differentiation

Take the derivative of the strength function with respect to \(h\): \(S' = 48 k h - 4 k h^{3}\). Setting this equal to zero to find the maximum(s), we get \(48 h - 4 h^{3} = 0\). Hence the value of \(h\) = 0 or ± √12. Discard negative and zero results, so \(h = √12 = 2 \sqrt{3}\), approx 3.46 inches.

Calculation of Width

Substituting \(h = 2 \sqrt{3}\) into \(w = \sqrt{576 - h^{2}}\), we have \(w = 2 \sqrt{12}\), approx 6.93 inches. Hence, the strongest beam that can be cut from the round log of diameter 24 inches has a height of approximately 3.46 inches and a width of approximately 6.93 inches.

Key concepts

Proportional Relationships

Understanding proportional relationships is fundamental in many areas of mathematics and everyday life. In the context of the exercise, the strength of a wooden beam is said to be directly proportional to its width and the square of its height. This means that as the width and square of the height increase, the strength increases at a consistent rate - this rate is represented by the proportionality constant, denoted by k. To visualize this, imagine you're adjusting sliders for width and height: sliding them up amplifies the beam's strength. This concept can be critical when we consider structural engineering, where the correct proportions can mean the difference between a stable building and an unsafe one.

Moreover, in our daily experiences, when we talk about driving speed, cooking recipes, or even the brightness of a light bulb, we're often referring to proportions. For instance, if a recipe calls for double the amount of flour for double the number of cake servings, that's a direct proportional relationship at play: cake servings to flour amount.

Pythagorean Theorem

The Pythagorean theorem is a principle that sets the foundation for geometry. It states that in a right-angled 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. This theorem is elegantly simple yet profoundly powerful and is used in our problem to relate the height and width of the rectangular beam to the diameter of the circular log. By visualizing the rectangular cross-section as the right triangle's legs, and the diameter as the hypotenuse, we can construct an equation that will connect the dimensions of the rectangle to the known diameter of the log. This application is not just theoretical; it has real-world implications in fields like construction, navigation, and design, where precise measurements are crucial.

Derivative Applications

Derivatives are a core component of calculus and have vast applications in various fields, such as economics, physics, and engineering. They represent the rate at which a function is changing at any given point and are used to find maxima and minima - in other words, optimization. For our wooden beam, we use derivatives to determine the point at which the strength function reaches its maximum value. By taking the derivative of the strength function with respect to height and setting it to zero, we're effectively finding the peak point of strength, akin to finding the top of a hill on a graph. This calculation is key in creating the strongest possible beam from the given log, and such optimization problems are omnipresent in real life, from maximizing profits in business to minimizing the time taken in travel.

Mathematical Optimization

Mathematical optimization is the process of finding the best solution from all feasible solutions. In the context of the wooden beam, this involves maximizing the strength of the beam given the constraint of the log's size. Optimization encompasses a variety of problems and techniques, with the objective being to make something as effective as possible under given conditions. It requires setting up an objective function, such as the strength of the beam S, and then adjusting variables within constraints, such as the log's diameter, to find the best possible outcome. Optimization is a critical tool in decision-making processes, be it in resource allocation, logistics, product design, or machine learning algorithms. Understanding how to set up and solve these problems can open doors to efficient and effective solutions in both academic and professional arenas.