Problem 57 - Algebra 1

Question

In Exercises \(56-58,\) use the following information. Bridge sections expand as the temperature goes up, so a small expansion gap is left between sections when a bridge is built. As the sections expand, the width of the gap gets smaller. Suppose that for some bridge the expansion gap is 16.8 millimeters wide at \(10^{\circ} \mathrm{C}\) and decreases by 0.37 millimeter for every \(1^{\circ} \mathrm{C}\) rise in temperature. The temperature rises to \(t^{\circ} \mathrm{C}\). Write expressions for the temperature rise, the decrease in the width of the gap, and the new width of the gap.

Step-by-Step Solution

Answer

Temperature rise = \(t - 10\), Decrease in width = \(0.37 \times (t - 10)\), New gap width = \(16.8 - 0.37 \times (t - 10)\)

Step 1: Expression for Temperature Rise

The temperature rise is simply given by the change in temperature from the initial temperature. So it is \(t - 10\).

Step 2: Expression for Decrease in Width

The decrease in the width of the gap is proportional to the temperature rise, with a constant of proportionality of 0.37 mm. So the decrease in width is given by \(0.37 \times (t - 10)\).

Step 3: Expression for New Width of Gap

The new width of the gap is the initial width of the gap minus the decrease in width due to temperature rise. So it is \(16.8 - 0.37 \times (t - 10)\).

Key Concepts

Algebraic Expressions

To start with, an algebraic expression is a way of representing numbers and operations in a mathematical form using symbols, letters, or numbers. In the context of thermal expansion scenarios, these expressions become crucial in modeling physical changes. In our exercise, we are dealing with a bridge's expansion gap, where algebraic expressions help in quantifying changes as temperatures rise.

The first algebraic expression is used to represent the temperature rise, given by \( t - 10 \). Here, \( t \) stands for the new temperature, while 10 is the initial temperature in degrees Celsius. This symbolically captures the change in temperature, providing a foundation for further calculations.

• Algebra is key in transforming real-world problems into solvable equations.
• Expressions like \( t - 10 \) simplify complex ideas to solvable components.

Understanding these fundamental expressions enables students to translate physical phenomena into mathematical language, thus making predictions and solving problems more manageable.

Linear Equations

In our bridge expansion exercise, we delve into linear equations, which are crucial for defining relationships with a constant rate of change. Linear equations assume the form \( y = mx + b \), where \( m \) is the slope or rate of change, and \( b \) is the y-intercept.

Considering the information given, the decrease in the width of the gap due to a rise in temperature is a classic example of a linear relationship. The decrease is calculated as \( 0.37 \times (t - 10) \), where 0.37 mm/°C is the rate of change, indicating how many millimeters the gap narrows per degree increase in temperature.

• Slope (\( m \)): Represents the rate of decrease per degree Celsius, here it is 0.37.
• The variable \( t-10 \): Reflects the temperature increase beyond the initial 10°C.

Linear equations enable precise calculations of changes based on initial conditions and rates, making them indispensable for modeling scenarios like thermal expansion.

Temperature Change Calculations

Temperature change calculations in algebra involve quantifying how varying temperatures influence other variables, like the width of a bridge gap. Understanding this allows engineers to predict structural behavior and plan accordingly to avoid potential problems.

From the original problem, we calculate the new width of the gap with a formula: \( 16.8 - 0.37 \times (t - 10) \).

• The original width of the gap at 10°C is 16.8 mm.
• Each degree rise in temperature shrinks the gap by 0.37 mm, realized through \( 0.37 \times (t - 10) \).

In practical terms, this calculation helps engineers ensure that structures like bridges can effectively accommodate temperature variations without compromising safety or functionality. Such calculations underpin the planning and maintenance of critical infrastructure, ensuring resilience to temperature-induced stress.