Problem 82 - Precalculus : Building Concepts and Connections

Question

Let \(T(x)\) be the temperature, in degrees Celsius, of a point on a long rod located \(x\) centimeters from one end of the rod (where that end of the rod corresponds to \(x=0\) ). Temperature can be measured in kelvin (the unit of temperature for the absolute temperature scale) by adding 273 to the temperature in degrees Celsius. Let \(t(x)\) be the temperature function in kelvin, and write an expression for \(t(x)\) in terms of the function \(T(x)\).

Step-by-Step Solution

Answer

The temperature in Kelvin at a certain distance, \(x\), from the end of the rod can be expressed in terms of the temperature in Celsius at the same point, \(T(x)\), by the relation \(t(x) = T(x) + 273\).

Step 1: Understand the task

The task is to write an expression for \(t(x)\), the temperature function in Kelvin, in terms of \(T(x)\), the temperature function in Celsius. To convert Celsius to Kelvin, we add 273.

Step 2: Express \(t(x)\) in terms of \(T(x)\)

Using the conversion rule given, we can represent the temperature in kelvin, \(t(x)\), as the temperature in Celsius, \(T(x)\), plus 273. So, \(t(x) = T(x)+273\).

Key Concepts

Celsius to Kelvin

Temperature is often measured in Celsius, but in scientific contexts, Kelvin is frequently used. The Kelvin scale is an absolute temperature scale that starts at absolute zero, making it ideal for scientific calculations and observations. To convert a temperature from Celsius to Kelvin, you simply add 273.15 to the Celsius figure. However, for simplicity in some contexts, we often just add 273. This adjustment accounts for the difference in the starting point of the two scales. While Celsius is based on the freezing and boiling points of water, Kelvin begins at absolute zero, the point where all thermal motion ceases.

Temperature Function

In the context of this exercise, we are dealing with a temperature function. A function like this can mathematically model how temperature varies along a rod. The function is denoted as \(T(x)\), where \(x\) indicates the position along the rod. This function gives us the temperature in degrees Celsius at any given point \(x\). Understanding this concept is crucial for working with functions that describe physical phenomena, as they allow us to predict and calculate changes in temperature over a specified distance.

Mathematical Expression

A mathematical expression is a combination of numbers, variables, and operations that define a specific value or relationship. Here, we want to express a new function, \(t(x)\), using an existing temperature function \(T(x)\). Since \(t(x)\) measures temperature in Kelvin, we derive it by using the conversion relationship for temperature: \(t(x) = T(x) + 273\). This expression is created by applying the conversion rule—adding 273 to the Celsius temperature, formulated as a simple mathematical addition.

Step by Step Solution

Let's break down how to find the desired expression, \(t(x)\). The problem involves understanding a conversion between Celsius and Kelvin, and applying it to a temperature function.

• Step 1: Identify what you're asked to accomplish. Here, it's converting a temperature from Celsius to Kelvin in the form of an algebraic expression.


• Step 2: Use the formula for conversion. You add 273 to the Celsius temperature. So, for our function, we convert \(T(x)\) to \(t(x)\), leading to the expression \(t(x) = T(x) + 273\).

This straightforward procedure shows the power of mathematical functions to handle various scenarios, like translating temperature measures along a rod.