Problem 18 - Technical Mathematics with Calculus

Question

The temperature \(T\) at a distance \(x\) in. from the end of a certain heated bar is given by \(T=2.24 x^{3}+1.85 x+95.4\left(^{\circ} \mathrm{F}\right) .\) Find the rate of change of temperature with respect to distance, which is called the temperature gradient, at a point 3.75 in. from the end.

Step-by-Step Solution

Answer

The temperature gradient at a point 3.75 in. from the end is T'(3.75) = 3 * 2.24 * (3.75)^2 + 1.85 degrees Fahrenheit per inch.

Step 1: Identify the Function and the Variable

First, recognize the temperature function given for the heated bar, which is in terms of the distance from the end: T(x) = 2.24x^3 + 1.85x + 95.4. Here, x represents the distance in inches from the end of the bar.

Step 2: Differentiate the Temperature Function

To find the rate of change of temperature with respect to distance, take the derivative of the temperature function T(x) with respect to x. The derivative, T'(x), gives us the temperature gradient: T'(x) = d/dx (2.24x^3 + 1.85x + 95.4) = 3 * 2.24x^2 + 1.85.

Step 3: Evaluate the Temperature Gradient at x = 3.75 in.

Now, substitute x = 3.75 into the derived temperature gradient function to find the rate of change at that particular distance: T'(3.75) = 3 * 2.24 * (3.75)^2 + 1.85. Then, calculate the value to get the temperature gradient at that point.

Key Concepts

Differentiation

Imagine you're driving a car and your friend asks you how fast you're going. You glance at the speedometer and give an answer: that's the concept of differentiation in a nutshell. It's about finding the instantaneous 'speed' or rate at which something is changing.

In the context of calculus, differentiation is the process of determining the rate at which a function is changing at any point. For our temperature function governing the heated bar, the math behind it is a bit more detailed. You differentiate the function by applying rules that break down how the temperature varies with respect to changes in distance. Doing so leads to expressions involving powers of the variable, which for our example is distance 'x'.

In the given exercise, the differentiation of the temperature function, written mathematically as the derivative, T'(x), shows how rapidly the temperature changes as you move along the bar. It allows us to find out if the bar is getting hotter or cooler as one moves away from the end, which is an essential piece of understanding not just temperature distribution but also concepts in physics, engineering, and beyond.

Rate of Change

Tied closely to the idea of differentiation is the rate of change. It hones in on the speed of change in one variable relative to another. For example, when you press down on your car's accelerator pedal, your car's speed changes. The rate of change is how quickly that speed increases per second.

In our heated bar scenario, the rate of change refers to the temperature gradient — specifically, it's how much the temperature changes for a tiny increment in distance along the bar. When you differentiate the temperature function with respect to distance, you're finding a formula to calculate this rate of change at any point on the bar.

The importance of understanding rate of change spans economics, sciences, and mathematics. It provides insights into trends and can predict future behavior of a system - critical information whether you're looking at stock prices, chemical reactions, or, as in our case, temperature distributions.

Temperature Function

Finally, let's focus on the temperature function itself. A temperature function is a practical example of how math models real-world phenomena. In this case, it maps the temperature along a bar to the distance from the bar's end. Like the equation you might see on a weather chart predicting temperature changes over the course of a day, the temperature function gives us a precise way to describe and predict heat distribution.

In the provided exercise, the function incorporates a cubic term, a linear term, and a constant. The cubic term suggests that as we move along the bar, temperature changes at a rate that's not constant but depends on the cube of the distance. This might correspond to the physical properties of the bar or the way it's heated.

The beauty of a temperature function, and indeed any function modeling a natural process, is that it turns qualitative descriptions into quantitative predictions. With T(x), we go from saying 'the bar gets hotter as you move closer to the end' to 'at 3.75 inches from the end, the temperature is rising at this specific rate,' which is a much more powerful and useful statement.