Problem 44 - Thomas Calculus in SI Units

Question

The temperature of an ingot of silver is \(60^{\circ} \mathrm{C}\) above room temperature right now. Twenty minutes ago, it was \(70^{\circ} \mathrm{C}\) above room temperature. How far above room temperature will the silver be a. 15 min from now? b. 2 hours from now? c. When will the silver be \(10^{\circ} \mathrm{C}\) above room temperature?

Step-by-Step Solution

Answer

a. 52.5°C b. 0°C (room temperature) c. In 100 minutes.

Step 1: Determine the initial temperature difference

We know that 20 minutes ago, the temperature of the silver was 70°C above room temperature. Right now, it's 60°C above room temperature. This means the temperature difference has decreased by 10°C in 20 minutes.

Step 2: Calculate the rate of temperature change

Given that the temperature has decreased by 10°C over 20 minutes, the rate of temperature change is \[ \text{Rate} = \frac{10^{\circ}\text{C}}{20\, \text{minutes}} = 0.5^{\circ}\text{C per minute} \]

Step 3: Calculate the temperature difference 15 minutes from now

Continuing at the same rate of cooling, in the next 15 minutes the temperature will decrease by:\[ 15 \times 0.5^{\circ}\text{C} = 7.5^{\circ}\text{C} \]Since the current temperature difference is 60°C, it will be:\[ 60^{\circ}\text{C} - 7.5^{\circ}\text{C} = 52.5^{\circ}\text{C} \] above room temperature.

Step 4: Compute the temperature difference 2 hours from now

Two hours is 120 minutes. Over this time, the temperature decrease will be:\[ 120 \times 0.5^{\circ}\text{C} = 60^{\circ}\text{C} \]Starting from the current temperature difference of 60°C, it will be:\[ 60^{\circ}\text{C} - 60^{\circ}\text{C} = 0^{\circ}\text{C} \] above room temperature. The silver will reach room temperature.

Step 5: Find when the temperature is 10°C above room temperature

To find out when the silver will be 10°C above room temperature, calculate how long it takes to cool from 60°C to 10°C. The decrease needed is:\[ 60^{\circ}\text{C} - 10^{\circ}\text{C} = 50^{\circ}\text{C} \]With a rate of 0.5°C per minute, the time required is:\[ \frac{50^{\circ}\text{C}}{0.5^{\circ}\text{C per minute}} = 100\, \text{minutes} \]Thus, in 100 minutes from now, the silver will be 10°C above room temperature.

Key Concepts

Cooling Rate Calculation

When discussing the rate at which an object cools down, we often refer to this as the 'cooling rate'. In the exercise, the goal was to determine how quickly the silver temperature changes over time. So, how do we figure this out?
The first step is to identify the change in temperature over a given time period. Here, twenty minutes ago, the silver was at a higher temperature difference than it is now. It cooled by 10°C in those 20 minutes. From this information, we can calculate the cooling rate by dividing the temperature change by the time period:

• Temperature change: 10°C
• Time period: 20 minutes

Thus, we obtain a cooling rate of 0.5°C per minute, which essentially tells us how much cooler the silver gets every minute. Understanding this concept is crucial as it allows us to predict future temperatures based on the rate of change.

Temperature Difference

Temperature difference is a key concept when analyzing how far a current temperature is from a reference point, such as room temperature. In this situation, the silver's temperature is consistently being compared to the room temperature.
Temperature difference can change over time as the object gains or loses heat. For example, at the start of this exercise, the temperature was 60°C above room temperature. Calculating future temperature differences involves applying the cooling rate accordingly.
For instance, 15 minutes later, we know that the temperature will drop by:

• Cooling Rate: 0.5°C per minute x 15 minutes = 7.5°C

So, the new temperature difference would be:

• Previous Difference: 60°C - 7.5°C = 52.5°C

These calculations help us understand how temperature differences evolve over time based on initial conditions and cooling rates.

Time-Temperature Relationship

Understanding the time-temperature relationship is essential when predicting how long it will take for an object to cool or heat to a certain degree above or below a set baseline, such as room temperature. The rate of temperature change, in this instance, helps us forecast future temperatures.
As outlined in the problem, the method involves using the known cooling rate and current temperature difference to estimate how the temperature will change over time. For example, if we need to find when the silver will be at 10°C above room temperature, we calculate the required decrease from the current 60°C to 10°C:

• Required Decrease: 60°C - 10°C = 50°C

With the cooling rate known (0.5°C per minute), the time can be computed as:

• Time Needed: 50°C ÷ 0.5°C per minute = 100 minutes

This relationship provides a direct way to determine when the desired temperature difference will be reached.