Problem 26 - Algebra A Combined Function

Question

The maximum weight that a circular column can hold is inversely proportional to the square of its height. If an 8 -foot column can hold 2 tons, find how much weight a 10 -foot column can hold.

Step-by-Step Solution

Answer

A 10-foot column can hold 1.28 tons.

Step 1: Understand Inverse Proportionality

The problem states that the weight a column can hold, \( W \), is inversely proportional to the square of its height, \( h \). This means we can express this relationship as \( W = \frac{k}{h^2} \), where \( k \) is a constant of proportionality that we'll need to determine.

Step 2: Find the Constant of Proportionality

To find \( k \), use the information provided: an 8-foot column can hold 2 tons. Plug these values into the relationship \( W = \frac{k}{h^2} \), giving: \( 2 = \frac{k}{8^2} \). Simplifying, \( 2 = \frac{k}{64} \), which leads to \( k = 2 \times 64 = 128 \).

Step 3: Calculate the Weight a 10-foot Column Can Hold

Now we have the formula \( W = \frac{128}{h^2} \). Substitute \( h = 10 \) to find the weight a 10-foot column can hold: \( W = \frac{128}{10^2} = \frac{128}{100} = 1.28 \) tons.

Key Concepts

Constants of Proportionality

In problems involving inverse proportionality, the constant of proportionality, denoted by \( k \), plays a crucial role. In our exercise, the weight a circular column can support, \( W \), is inversely proportional to the square of its height, \( h \). We express this relationship mathematically as \( W = \frac{k}{h^2} \). Here, the constant \( k \) is a specific value that relates the two variables in this equation.

To determine the constant \( k \), we need a reference point — a known weight and height. From the exercise, we know that an 8-foot column can support 2 tons. Plugging these values into the equation, we find \( k \) as follows: \( 2 = \frac{k}{8^2} = \frac{k}{64} \). Solving for \( k \), we get \( k = 2 \times 64 = 128 \). This constant not only helps solve the problem but also serves as a foundation for understanding how changes in height affect the column's capacity to support weight.

Mathematical Modeling

Mathematical modeling is a process where real-world problems are represented using mathematical concepts and language. In our exercise, we modeled the relationship between the column's height and the weight it can support using an inverse proportionality formula. This mathematical approach is crucial because it allows us to predict outcomes under different conditions and make informed decisions.

The formula \( W = \frac{k}{h^2} \) highlights this modeling. Here, the relationship is not direct but inverse, meaning as one value increases, the other decreases. Through solving, we can adapt the model to respond to changes like height modifications. When height increases, the weight holding capacity decreases, as demonstrated by calculating the weight a 10-foot column can support.

Mathematical models like this one are fundamental in engineering, physics, economics, and many other fields. They simplify complex relationships into manageable formulas, providing a framework to analyze and predict behavior.

Problem-solving in Algebra

Algebraic problem-solving involves translating given data into mathematical expressions and equations to find unknown quantities. Let's break down the steps in this exercise to illustrate this approach.

• First, understand the type of relationship present: here, it's inverse proportionality.
• Next, establish the formula that represents this relationship. We used \( W = \frac{k}{h^2} \) from the problem statement.
• Determine any constants by substituting known values. For example, with an 8-foot column holding 2 tons, we found \( k = 128 \).
• Finally, solve for the unknown by substituting the new conditions into the established formula. For a 10-foot column, substituting into \( W = \frac{128}{h^2} \) gives us the weight capacity as 1.28 tons.

Algebra enables a structured approach to solving problems by reducing complex scenarios to simple equations. This structured approach is powerful in tackling a wide variety of real-world problems and scenarios effectively.