Problem 29 - Calculus

Question

Write a formula representing the function. The strength, \(S\), of a beam is proportional to the square of its thickness, \(h\)

Step-by-Step Solution

Answer

The formula is \( S = k h^2 \).

Step 1: Identify the Variables

We are given that the strength of the beam is denoted by \( S \) and the thickness of the beam is denoted by \( h \).

Step 2: Understand the Relationship

The problem states that the strength \( S \) is proportional to the square of the thickness \( h \). In mathematical terms, this means \( S \propto h^2 \).

Step 3: Convert Proportionality to an Equation

We write the proportional relationship as an equation by introducing a constant of proportionality \( k \). This results in the equation: \[ S = k h^2 \] where \( k \) is a constant.

Step 4: Conclusion

The formula that represents the relationship between the strength \( S \) and the thickness \( h \) of the beam is \[ S = k h^2 \]. This formula expresses that \( S \) is directly proportional to the square of \( h \) with proportion constant \( k \).

Key Concepts

Mathematical Formula

A mathematical formula is a concise way of expressing information symbolically, as in an equation. It gives us a clear and precise method to calculate values or outcomes based on given conditions and relationships.

In the context of our exercise, we are dealing with a relationship where the strength of a beam, denoted by \( S \), is dependent on the thickness of the beam, represented by \( h \). The relationship is such that the strength varies according to the square of the thickness. To translate this verbal relationship into mathematical terms, we use a proportional relationship. This can be expressed in the proportionality form \( S \propto h^2 \).

Mathematical formulas like \( S = k h^2 \) allow us to calculate the strength of the beam if the thickness and the constant \( k \) are known. Formulas are tools that help mathematicians and engineers alike to predict and understand complex systems in a straightforward manner.

Direct Proportionality

Direct proportionality describes a relationship between two quantities where they increase or decrease in tandem. If one variable increases, the other one increases at a constant rate and vice versa. This is expressed with mathematical notation using the symbol \( \propto \).

In simpler terms, if we say that something, like the strength \( S \), is directly proportional to another variable \( h^2 \) (here, the square of thickness), it means that as \( h^2 \) grows, \( S \) grows too, and if \( h^2 \) decreases, \( S \) decreases. The relationship implies a steady rate of change between these variables.

For our beam exercise, the relationship \( S \propto h^2 \) indicates that the strength of the beam will consistently change with the square of its thickness. This relationship is pivotal because it allows predictions about the strength based on thickness changes.

Constant of Proportionality

The constant of proportionality, often denoted by \( k \), is a fixed value used in proportional relationships to convert a proportionality statement into an equation. It serves two main purposes: to complete the equation and to provide the scale or weight of the proportional relationship.

In the equation \( S = k h^2 \), \( k \) acts as the bridge that completes the process from a proportional statement \( S \propto h^2 \) to a precise equation. Without \( k \), we would only know that there is some relationship between \( S \) and \( h^2 \), but not the specific strength for a given thickness.

Finding the value of \( k \) typically involves using empirical data or measurements. Once known, \( k \) allows for accurate calculations of one variable when the other is known, making it a crucial component in engineering and physics calculations, such as analyzing beam strength in this exercise.