Problem 51
Question
Temperature change It took 14 sec for a mercury thermometer to rise from \(-19^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) when it was taken from a freezer and placed in boiling water. Show that somewhere along the way the mercury was rising at the rate of \(8.5^{\circ} \mathrm{C} / \mathrm{sec}\).
Step-by-Step Solution
Verified Answer
The rate was \(8.5^{\circ} \mathrm{C/sec}\) somewhere during the 14 seconds.
1Step 1: Understand the Average Rate of Temperature Change
The thermometer went from \(-19^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) in 14 seconds. The average rate of change in temperature can be calculated using the formula for average rate of change: \[ \text{Average Rate} = \frac{\text{Change in Temperature}}{\text{Time}} \]
2Step 2: Calculate Change in Temperature
The change in temperature is the final temperature minus the initial temperature: \[ \text{Change in Temperature} = 100^{\circ} \mathrm{C} - (-19^{\circ} \mathrm{C}) = 100^{\circ} \mathrm{C} + 19^{\circ} \mathrm{C} = 119^{\circ} \mathrm{C} \]
3Step 3: Substitute Values into Average Rate Formula
Substitute the change in temperature and the time into the formula from Step 1: \[ \text{Average Rate} = \frac{119^{\circ} \mathrm{C}}{14 \text{ sec}} \approx 8.5^{\circ} \mathrm{C/sec} \]
4Step 4: Apply the Mean Value Theorem
According to the Mean Value Theorem for derivatives, there exists at least one point in the interval where the instantaneous rate of change (derivative) is equal to the average rate of change. This means there must be a point where the thermometer's rate matched the calculated average of \(8.5^{\circ} \mathrm{C/sec}\).
Key Concepts
Average Rate of ChangeInstantaneous Rate of ChangeTemperature Change
Average Rate of Change
The average rate of change is a fundamental concept when analyzing how something varies over a specific period. It's like looking at a journey and figuring out the overall speed, regardless of any temporary stops or changes in pace during the trip. For temperature scenarios, such as with our thermometer, the average rate of change helps us know how quickly or slowly the temperature increased overall.
Let's consider our original problem: a thermometer moves from \(-19^{\circ}\text{C}\) to \(100^{\circ}\text{C}\) over 14 seconds. The average rate of change is calculated by dividing the total change in temperature by the time it took. So, the formula looks like:
Let's consider our original problem: a thermometer moves from \(-19^{\circ}\text{C}\) to \(100^{\circ}\text{C}\) over 14 seconds. The average rate of change is calculated by dividing the total change in temperature by the time it took. So, the formula looks like:
- Average Rate = \(\frac{\text{Change in Temperature}}{\text{Time}}\)
- Change in Temperature = \(100^{\circ}\text{C} - (-19^{\circ}\text{C}) = 119^{\circ}\text{C}\)
- Average Rate = \(\frac{119^{\circ}\text{C}}{14\text{ sec}} \approx 8.5^{\circ}\text{C/sec}\)
Instantaneous Rate of Change
In contrast to the average rate of change, the instantaneous rate of change tries to pinpoint exactly how fast something is changing at a single moment, rather than over a period. In calculus, this is where derivatives come into play. It provides us with a precise speed or rate at any given second.
For our thermometer example, the Mean Value Theorem gives insight. This theorem tells us that within our 14-second period, there must be at least one instant where the thermometer's rate of temperature increase is exactly the average of \(8.5^{\circ}\text{C/sec}\).
It's like driving a car on a highway; even if your average speed is 55 mph throughout the journey, there was some point where you were going exactly 55 mph — even if you sped up or slowed down other times. The Mean Value Theorem acts as a bridge connecting an overall journey to the precise speed at least once along the path.
For our thermometer example, the Mean Value Theorem gives insight. This theorem tells us that within our 14-second period, there must be at least one instant where the thermometer's rate of temperature increase is exactly the average of \(8.5^{\circ}\text{C/sec}\).
It's like driving a car on a highway; even if your average speed is 55 mph throughout the journey, there was some point where you were going exactly 55 mph — even if you sped up or slowed down other times. The Mean Value Theorem acts as a bridge connecting an overall journey to the precise speed at least once along the path.
Temperature Change
Temperature changes are often described using rates like average and instantaneous change because they provide clear, quantitative insights into how temperatures behave over time, especially in dynamic environments such as freezing or boiling scenarios.
In real-world applications, understanding temperature change is crucial. This could be in scientific experiments where precise thermal control is needed, or in everyday situations such as cooking or climate control systems. Temperature change impacts a multitude of areas, from food safety to weather prediction.
In our exercise, the thermometer's jump from \(-19^{\circ}\text{C}\) to \(100^{\circ}\text{C}\) not only highlights how rapidly temperatures can shift but also the efficiency of the measuring tool itself. Tracking temperature changes at different rates helps us predict, manage, and understand various processes related to heat and temperature.
In real-world applications, understanding temperature change is crucial. This could be in scientific experiments where precise thermal control is needed, or in everyday situations such as cooking or climate control systems. Temperature change impacts a multitude of areas, from food safety to weather prediction.
In our exercise, the thermometer's jump from \(-19^{\circ}\text{C}\) to \(100^{\circ}\text{C}\) not only highlights how rapidly temperatures can shift but also the efficiency of the measuring tool itself. Tracking temperature changes at different rates helps us predict, manage, and understand various processes related to heat and temperature.
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