Problem 56 - 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. If the temperature is \(18^{\circ} \mathrm{C},\) by how many degrees did the temperature rise? By how much would the width of the gap decrease? What would the new width of the gap be? Round to the nearest tenth of a millimeter.

Step-by-Step Solution

Answer

The temperature rose by 8 degrees. The width of the gap decreased by approximately 2.96 millimeters. The new width of the gap, rounded to the nearest tenth of a millimeter, is 13.8 millimeters.

Step 1: Identify the Given Information

From the exercise, we find that the initial temperature was \(10^{\circ}C\), and the corresponding gap width was 16.8 millimeters. Additionally, it's stated that the gap decreases by 0.37 millimeters for each degree rise in temperature. The current temperature is \(18^{\circ}C\).

Step 2: Determine the Rise in Temperature

To find out how many degrees the temperature rose, subtract the initial temperature from the current temperature: \(18^{\circ}C - 10^{\circ}C = 8^{\circ}C\). This means the temperature rose by 8 degrees.

Step 3: Determine the Change in the Gap Width

Now, to calculate the change in the gap size, multiply the change in temperature by the known rate of decrease in the gap size (0.37 millimeters per degree): \(8^{\circ}C * 0.37\,mm/^{\circ}C = 2.96\,mm\). That indicates the width of the gap decreased by approximately 2.96 millimeters ensuing from the temperature rise.

Step 4: Determine the New Width of the Gap

Lastly, subtract the change in width from the initial width of the gap to determine the new width: \(16.8\,mm - 2.96\,mm = 13.84\,mm\). Rounded to the nearest tenth of a millimeter, the new width of the gap would be 13.8 millimeters.

Key Concepts

Temperature Change

A change in temperature is when the temperature of an object or environment increases or decreases. It's essential to know how to calculate it when you are dealing with real-world applications, like the expansion of bridge sections. In this exercise, we start with an initial temperature of \(10^\circ C\) and see it increase to \(18^\circ C\).

To find the rise in temperature, simply subtract the starting temperature from the ending temperature:

• Current Temperature: \(18^\circ C\)
• Initial Temperature: \(10^\circ C\)

So, the temperature change is: \[18^\circ C - 10^\circ C = 8^\circ C\]
This straightforward calculation tells us just how much the temperature has increased.

Expansion Gap

An expansion gap is left between bridge sections when they are built. This gap allows the bridge to expand and contract with temperature changes without causing any structural damage. As the temperature increases, the materials of the bridge expand, and this gap becomes smaller.

In the bridge's initial state, at \(10^\circ C\), the expansion gap is \(16.8\) millimeters wide. However, every \(1\degree C\) rise in temperature reduces this gap by \(0.37\) millimeters.

With the temperature rising from \(10^\circ C\) to \(18^\circ C\), the total decrease in the gap can be calculated by multiplying the temperature rise by the gap's rate of decrease: \[8^\circ C \times 0.37\, \text{mm/}^\circ C = 2.96\, \text{mm}\]
This shows that the gap reduces by \(2.96\) millimeters due to the temperature increase.

Rate of Decrease

The rate of decrease is vital in understanding how quickly or slowly a measurement changes. In this situation, temperature rise directly affects how much the expansion gap shrinks. For every \(1^\circ C\) increase in temperature, the gap decreases by \(0.37\) millimeters.

Understanding the rate of decrease helps us predict and calculate changes effectively.

• Rate of decrease per degree: \(0.37\, \text{mm/}^\circ C\)

With an \(8^\circ C\) increase in temperature: \[8^\circ C \times 0.37\, \text{mm/}^\circ C = 2.96\, \text{mm}\]
Therefore, by understanding and utilizing this linear relationship, the gap decreases by \(2.96\) millimeters.