Question

A steel forging is \(1495^{\circ} \mathrm{F}\) above room temperature. If it cools exponentially at the rate of \(2.00 \%\) per minute, how much will its temperature drop in \(1 \mathrm{h} ?\)

Short answer

The drop in temperature after one hour can be calculated using the formula \(T(60) = 1495 \times (1 - 0.02)^{60}\), then finding the temperature difference between the initial state and after one hour.

Step-by-step solution

Understanding the Exponential Decay Formula

To solve the problem, we need to apply the formula for exponential decay, which is given by: Temperature after time t: \(T(t) = T_0 \times (1 - r)^t\), where:- \(T(t)\) is the temperature after time \(t\),- \(T_0\) is the initial temperature (the amount above room temperature in this case),- \(r\) is the decay rate per time unit,- \(t\) is the time in the same units as the decay rate.

Convert Time to Minutes

Since the decay rate is given per minute, we should convert the time from hours to minutes. There are 60 minutes in an hour, thus 1 hour is equal to 60 minutes.

Insert Given Values into the Formula

We now input the given values into the exponential decay formula. The initial temperature \(T_0\) is 1495°F above room temperature, the decay rate \(r\) is 2% per minute (0.02 as a decimal), and the time \(t\) is 60 minutes.So, \(T(60) = 1495 \times (1 - 0.02)^{60}\).

Calculate the Temperature Drop

Perform the calculation to find the temperature after 60 minutes. Use a calculator to find the value of \((1 - 0.02)^{60}\) and then multiply this by 1495°F.

Find the Temperature Drop

Subtract the final temperature from the initial temperature above room temperature to find the drop in temperature: \(\Delta T = T_0 - T(60)\).

Key concepts

Exponential Decay Formula

Understanding the exponential decay formula is crucial when analyzing situations where quantities decrease at a consistent rate over time. This formula, expressed as
\[ T(t) = T_0 \times (1 - r)^t \]
is used to model various real-world processes, including radioactive decay, population decrease, and, as in our exercise, the cooling of materials.

The variables in the formula represent the following: \( T(t) \) is the quantity at time \( t \); \( T_0 \) is the initial quantity; \( r \) is the decay rate as a decimal; and \( t \) is the elapsed time. The rate, \( r \) is typically given as a percentage and must be converted to a decimal by dividing by 100. For example, a 2% decay rate becomes 0.02 in the formula.

In this exercise, we're using the exponential decay formula to determine how much a steel forging cools over time. The initial temperature is given as 1495°F above room temperature and has a 2% decay per minute. To find out how much the temperature drops after one hour, the time \( t \) should be converted to minutes, since the given decay rate is per minute. Then, we can use these values to calculate \( T(t) \) and determine the amount of cooling.

Temperature Conversion

Temperature conversion is an essential skill when working with thermal systems, especially when comparing temperatures in different units, such as Fahrenheit (°F) and Celsius (°C). However, in our current exercise, the temperature is given in Fahrenheit and does not require unit conversion. But it is still helpful to understand how to switch between Fahrenheit and Celsius if needed:

\[ \text{To convert } °F \text{ to } °C: °C = \frac{5}{9} \times (°F - 32) \]
\[ \text{To convert } °C \text{ to } °F: °F = \frac{9}{5} \times °C + 32 \]
It is important to keep in mind that temperature differences can be converted using the same factor without needing to adjust for the different zero points of the scales. For instance, a temperature drop of 1°F is equivalent to a drop of \( \frac{5}{9} \)°C. This remains true regardless of the starting temperatures.

Rate of Cooling

The rate of cooling refers to how quickly the temperature of an object decreases over time. In our scenario with the cooling of a steel forging, we're dealing with an object that cools according to an exponential decay model. The rate at which this cooling occurs can be influenced by various factors, including the properties of the material, the surrounding environment, and the difference between the object's temperature and the ambient temperature.

In more technical terms, Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature. However, for certain materials and under specific conditions, the process simplifies to an exponential decay of temperature.

In the given exercise, we know the initial temperature difference between the steel forging and the room temperature is 1495°F and that the rate of cooling is 2% per minute. Using the exponential decay formula, one can calculate the temperature of the steel forging after any number of minutes to determine how rapidly it is losing heat. This understanding helps us predict and control cooling processes in various scientific and industrial applications.