Question

At the start of a dog sled race in Anchorage, Alaska, the temperature was \(5^{\circ} \mathrm{C}\). By the end of the race, the temperature was \(-10^{\circ} \mathrm{C}\). The formula for converting temperatures from degrees Fahrenheit \(F\) to degrees Celsius \(C\) is \(C=\frac{5}{9}(F-32)\). a. Find the inverse function. Describe what it represents. b. Find the Fahrenheit temperatures at the start and end of the race. c. Use a graphing calculator to graph the original function and its inverse. Find the temperature that is the same on both temperature scales.

Short answer

The inverse function is \(F = \frac{9}{5}C + 32\), which represents converting Celsius to Fahrenheit. The temperature at the start of the race is \(41^{\circ}F\) and at the end is \(14^{\circ}F\). The point of intersection on the graph of the original and inverse function indicates the temperature where Fahrenheit equals Celsius.

Step-by-step solution

Finding the Inverse Function

Given function is \(C=\frac{5}{9}(F-32)\). To find the inverse, first solve for \(F\). Multiply both sides by 9/5 to get \(F = \frac{9}{5}C + 32\). This inverse function represents converting temperature from Celsius to Fahrenheit.

Temperature Conversion at Start and End of Race

Substitute the temperatures at the start and end of the race into the inverse function to calculate Fahrenheit temperatures. For start of the race: \(F = \frac{9}{5}(5) + 32 = 41^{\circ}F\). For end of the race: \(F = \frac{9}{5}(-10) + 32 = 14^{\circ}F\)

Graphing Original and Inverse Function

You need to graph the original and inverse function using a graphing tool or graphing calculator. The point of intersection indicates the temperature that is the same on both scales. That's because at this point, the temperature in Fahrenheit equals the temperature in Celsius.

Key concepts

Temperature Conversion

Understanding temperature conversion is crucial when you need to relate temperature between different scales, like Celsius and Fahrenheit. The given exercise introduces the formula to convert Fahrenheit (denoted as \( F \)) to Celsius (\( C \)) which is:

• \( C = \frac{5}{9}(F - 32) \)

This allows you to take a temperature in Fahrenheit and find its equivalent in Celsius, which is useful when dealing with scientific data, travel, or even daily weather updates. Converting back, from Celsius to Fahrenheit, involves using the inverse function, as it's like reverse engineering the same relationship.
To find this inverse function, you flip the variables. So for \( F \), you solve the equation:

• \( F = \frac{9}{5}C + 32 \)

This formula indicates how you can convert a temperature from degrees Celsius to degrees Fahrenheit.

Function Graphing

Graphing functions is a helpful tool to visualize relationships between variables. In math, a function graph represents a set of ordered pairs and can show you how one variable changes with another. For instance, in this exercise, you were asked to graph both the original Fahrenheit to Celsius conversion function and its inverse.
Using a graphing calculator is a modern way to tackle such tasks effectively. Simply plot two equations on the graph:

• The original function: \( C = \frac{5}{9}(F - 32) \)
• The inverse function: \( F = \frac{9}{5}C + 32 \)

The point where these two graphs intersect is particularly intriguing as it shows the temperature which is the same on both scales. This point of intersection serves as a practical insight into how different units can coincide at certain values.

Fahrenheit to Celsius

Moving between Fahrenheit and Celsius scales is common and important for various scientific and everyday purposes. To start converting, remember the formula for converting Fahrenheit to Celsius is:

• \( C = \frac{5}{9}(F - 32) \)

This exercise provided Celsius temperatures at the start and end of a race in Alaska, and by using the inverse conversion, you were able to find the equivalent Fahrenheit values:

• Start of the race: \( F = \frac{9}{5}(5) + 32 = 41^{\circ}F \)
• End of the race: \( F = \frac{9}{5}(-10) + 32 = 14^{\circ}F \)

So, by using both direct and inverse formulas, you can smoothly convert measurements in either direction. This systematic approach ensures you can handle temperature readings in any context with confidence.