Question

A metal casting is placed in an environment maintained at a constant temperature, \(S_{0}\). Assume the temperature of the casting varies according to Newton's law of cooling. A thermal probe attached to the casting records the temperature \(\theta(t)\) listed. Use this information to determine (a) the initial temperature of the casting. (b) the temperature of the surroundings. $$\theta(t)=70+270 e^{-t}{ }^{\circ} \mathrm{F}$$

Short answer

Answer: (a) The initial temperature of the casting is \(340^\circ \mathrm{F}\) and (b) the temperature of the surroundings is \(70^\circ \mathrm{F}\).

Step-by-step solution

Evaluate the initial temperature

To find the initial temperature of the casting, we need to evaluate the temperature function, \(\theta(t)=70+270 e^{-t}\), at t=0. Substitute t=0 into the function: $$ \theta(0) = 70 + 270e^{-0} $$ Since \(e^0=1\), we have $$ \theta(0) = 70 + 270(1) = 70 + 270 = 340 $$

Determine the initial temperature

The initial temperature of the casting is the value of \(\theta(0)\), which we calculated in the previous step: $$ \theta_{initial} = 340^\circ \mathrm{F} $$

Evaluate the limit of the temperature function as t approaches infinity

To determine the temperature of the surroundings, take the limit of the temperature function as t approaches infinity: $$ \lim_{t \to \infty} \theta(t) = \lim_{t \to \infty} (70 + 270 e^{-t}) $$ Now, as t approaches infinity, the value of \(e^{-t}\) approaches 0. Thus, we have: $$ \lim_{t \to \infty} \theta(t) = 70 + 270(0) = 70 $$

Determine the temperature of the surroundings

The temperature of the surroundings is the limit of the temperature function as t approaches infinity, which we calculated in the previous step: $$ S_0 = 70^\circ \mathrm{F} $$ Therefore, (a) the initial temperature of the casting is \(340^\circ \mathrm{F}\), and (b) the temperature of the surroundings is \(70^\circ \mathrm{F}\).

Key concepts

Initial Temperature Calculation

Understanding the initial temperature of a system is crucial when studying thermal dynamics. The initial temperature is essentially the starting point from where a body begins to change its temperature due to its surroundings. In the context of Newton's law of cooling, the initial temperature is determined by evaluating the temperature function at the time when the observation begins, typically when time, t, equals zero.

For the given exercise, the temperature function is \(\theta(t) = 70 + 270 e^{-t}\). By substituting t with zero, we simplified the expression to \(\theta(0) = 340^\circ \mathrm{F}\). This means that at the starting time, which is when the metal casting was freshly placed in the environment, its temperature was 340 degrees Fahrenheit.

Temperature Function

A temperature function models how the temperature of an object changes over time. Newton's law of cooling describes this change as a process where the rate of heat loss of the object is proportional to the difference between the object's temperature and the ambient temperature of the surroundings.

In the provided problem, the temperature function \(\theta(t)\) takes the form of \(70 + 270 e^{-t}\). The e in the equation stands for Euler's number, which is an important constant in mathematics. The exponential component \(e^{-t}\) depicts how the temperature changes based on time, indicating an exponential decrease in difference between the casting’s temperature and the environmental temperature over time.

Temperature of Surroundings

The temperature of the surroundings, often referred to as ambient temperature, plays a significant role in how an object cools or heats. In our case, the metal casting's temperature changes in response to the temperature of its environment, which is maintained at a constant \(S_0\).

To find \(S_0\), we evaluate the limit of the temperature function as time \(t\) approaches infinity. This is because, over time, the temperature of the casting will approach that of its surroundings. From the step-by-step solution, we determined that \(S_0\) equals 70 degrees Fahrenheit by evaluating \(\lim_{t \to \infty} \theta(t)\).

Limit of Temperature Function

The limit of a temperature function as time t approaches infinity provides us with valuable information: it indicates the steady-state temperature that the object will eventually achieve if the conditions remain constant. The steady-state temperature is the temperature at which the rate of heat transfer to the surroundings equals the rate at which the object is gaining heat, resulting in a temperature that no longer changes with time.

In our exercise, calculation of the limit showed that as time goes to infinity, the metal casting’s temperature would settle at the surrounding temperature of 70 degrees Fahrenheit. Mathematically, this was shown as \(\lim_{t \to \infty} \theta(t) = 70^\circ \mathrm{F}\), signifying the point when the temperature function \(\theta(t)\) no longer changes with time, which is key to understanding the final steady-state scenario according to Newton’s law of cooling.