Question

A civil engineer relates the thickness \(T,\) in inches, and height \(H,\) in feet, of a square wooden pillar to its crushing load \(L\), in tons, using the model \(T=\sqrt[4]{\frac{L H^{2}}{25}}\). If a square wooden pillar is 4 inches thick and 10 feet high, what is its crushing load?

Short answer

The crushing load is 64 tons.

Step-by-step solution

- Identify Given Values

Identify the given values in the problem. We have the thickness of the pillar, which is given as \( T = 4 \) inches, and the height of the pillar, which is given as \( H = 10 \) feet.

- Write down the Model Equation

The model provided relates the thickness \( T \), height \( H \), and crushing load \( L \) with the equation \( T = \sqrt[4]{\frac{L H^{2}}{25}} \).

- Substitute Given Values into the Equation

Substitute \( T = 4 \) and \( H = 10 \) into the equation: \[ 4 = \sqrt[4]{\frac{L (10)^{2}}{25}} \]

- Simplify the Equation

Simplify the equation inside the fourth root: \[ 4 = \sqrt[4]{\frac{L \times 100}{25}} \] \[ 4 = \sqrt[4]{4L} \]

- Solve for L

Raise both sides of the equation to the power of 4 to remove the fourth root: \[ (4)^{4} = 4L \] Calculate \( (4)^{4} \): \[ 256 = 4L \] Divide both sides by 4 to solve for \( L \): \[ L = \frac{256}{4} = 64 \]

Key concepts

solving equations

Solving equations is a fundamental part of algebra and critical in various engineering applications. It involves finding the unknown values that satisfy an equation. In our problem, we need to solve for the crushing load, denoted as \( L \), using the provided model equation.

First, we identify all known values. We know the thickness of the pillar \( T = 4 \) inches and the height \( H = 10 \) feet.
Next, we use these values in our model equation: \( T = \sqrt[4]{\frac{L H^{2}}{25}} \). By substituting these values, we obtain: \( 4 = \sqrt[4]{\frac{L \times 100}{25}} \).

• Simplify the equation: \( 4 = \sqrt[4]{4L} \)
• Isolate the variable \( L \): \((4)^{4} = 4L\)
• Solve the resulting equation: \( 256 = 4L \)
• Finally, divide by 4: \( L = 64 \) tons

Thus, the crushing load \( L \) is 64 tons. This systematic approach, solving equations step-by-step, allows for resolving complex algebraic problems efficiently.

algebraic modeling

Algebraic modeling is the practice of using algebraic expressions to represent real-world scenarios. This is highly valuable in engineering, where predicting and understanding relationships between variables is critical.

In our exercise, we use the equation \( T = \sqrt[4]{\frac{L H^{2}}{25}} \) to model the relationship between the thickness \( T \), height \( H \), and crushing load \( L \) of a wooden pillar.

Through algebraic modeling, we:

• Translate physical phenomena into mathematical terms
• Utilize variables to represent different components, such as thickness (\( T \)), height (\( H \)), and load (\( L \))
• Use the model to predict outcomes, like solving for the unknown crushing load \( L \)

By substituting the known values of \( T \) and \( H \) into the equation, we simplify and solve for the unknown \( L \). Algebraic models like this help engineers design safer and more efficient structures.

applied mathematics

Applied mathematics involves using mathematical techniques to solve practical problems in everyday life, especially in engineering. It encompasses various fields including algebra, calculus, and statistics.

In the example, we used applied mathematics to determine the crushing load of a wooden pillar. This process involved:

• Recognizing the problem's real-world context
• Using algebraic rules to manipulate and solve the given equation
• Applying mathematical models to derive meaningful results

This type of problem-solving is crucial for engineers. It ensures that the structures they design are safe and reliable. Applied mathematics bridges theory and practice, enabling engineers to predict, test, and ultimately implement their designs effectively.
Understanding the principles behind the math not only aids in solving specific problems but also builds the foundation for tackling more complex engineering challenges in the future.