Problem 19 - Technical Mathematics with Calculus

Question

Cooling: A certain iron casting is at \(1800^{\circ} \mathrm{F}\) and cools so that its temperature at each minute is \(10 \%\) less than its temperature the preceding minute. Find its temperature after 1 h.

Step-by-Step Solution

Answer

The temperature of the iron casting after 1 hour is approximately 4.446°F.

Step 1: Understanding the Problem

The given problem is a cooling problem that involves exponential decay of temperature. The iron casting's temperature decreases by 10% each minute. We need to calculate the temperature after 60 minutes (1 hour) because there are 60 minutes in an hour.

Step 2: Setting up the Exponential Decay Formula

We can use the formula for exponential decay, which is \(T(t) = T_0 \times (1 - r)^t\), where \(T(t)\) is the temperature at time \(t\), \(T_0\) is the initial temperature, \(r\) is the decay rate (10% in this case, which is 0.10), and \(t\) is the time in minutes.

Step 3: Plugging in the Given Values

The initial temperature \(T_0\) is 1800°F, the decay rate \(r\) is 0.10, and the time \(t\) after which we want to find the temperature is 60 minutes. Substituting these values into the formula, we get \(T(60) = 1800 \times (1 - 0.10)^{60}\).

Step 4: Calculating the Temperature After 60 Minutes

Now, calculate the expression \((1 - 0.10)^{60}\) using a calculator, and then multiply the result by 1800 to find the temperature after 60 minutes. This will give us the temperature of the iron casting after 1 hour.

Step 5: Final Calculation

Using the calculator, find \((1 - 0.10)^{60}\) which equals approximately 0.00247. Multiply this by 1800 to obtain the temperature \(T(60) = 1800 \times 0.00247\).

Key Concepts

Cooling Problem Mathematics

When studying the cooling of objects, mathematics offers us a powerful tool through the concept of exponential decay. This concept models how the temperature of an object decreases over time. Firstly, it's important to understand that real-world cooling does not happen linearly; instead, temperatures typically drop more quickly when the difference between the object and its surroundings is greater, and slow down as this difference decreases.

For example, the iron casting in our exercise started at a high temperature and cools down in such a way that every minute its temperature is reduced by a fixed percentage. This behavior can be better understood by visualizing a curve that drops sharply at first and then levels off as time passes. By applying an exponential decay formula, we can predict the temperature at any given minute, thus solving what's known as a cooling problem in mathematics.

Temperature Decay Rate Calculation

Calculating the temperature decay rate is central to solving any cooling problem. In our scenario, the decay rate is given as 10% per minute. This percentage must be converted into a decimal to be used in calculations, becoming 0.10. The decay rate reflects the rate at which the temperature of an object is reduced relative to its current temperature.

One way to approach this is to think of the temperature as being multiplied by a factor less than one (in our case, 90% or 0.90) every time unit. Over time, these successive multiplications results in the temperature dropping towards but never actually reaching absolute zero, due to the nature of exponential decay. Accuracy in determining and applying the decay rate is crucial, as it dictates the rate at which the temperature changes, affecting the cooling curve significantly.

Exponential Decay Formula Application

The application of the exponential decay formula involves taking the initial value and applying the decay factor to this value repeatedly over the period of time in question.

In our example, the initial temperature, represented by \(T_0\), is 1800°F. The decay factor, \(1 - r\), is 0.90, since the decay rate \(r\) is 0.10. For every minute that passes, we multiply the temperature by 0.90. It’s like saying at every step, the temperature retains 90% of its value from the previous minute. After 60 minutes, we have to calculate \(0.90^{60}\). This calculation tells us by what factor the initial temperature has been reduced after one hour. Multiplying this factor by the initial temperature gives the final temperature, \(T(60)\), which would be substantially lower than 1800°F due to a compounded effect of the 10% decrease applied 60 times.