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Chapter 1: Problem 13

The relationship between the Fahrenheit \((F)\) and Celsius \((C)\) temperature scales is given by the linear function \(F=\frac{9}{5} C+32 .\) (a) Sketch a graph of this function. (b) What is the slope of the graph and what does it represent? What is the \(F\) -intercept and what does it represent?

Short Answer

Expert verified
The slope is \(\frac{9}{5}\) (each 1掳C increase results in 1.8掳F increase); F-intercept is 32掳F (temperature at 0掳C).

Step by step solution

01

Understand the Equation

The equation given is \( F = \frac{9}{5} C + 32 \). This is a linear equation in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( F \) represents the temperature in Fahrenheit and \( C \) the temperature in Celsius.
02

Identify the Slope and Intercept

From the equation \( F = \frac{9}{5} C + 32 \), the slope \( m \) is \( \frac{9}{5} \) and the y-intercept \( b \) is 32. The slope \( \frac{9}{5} \) represents the increase in Fahrenheit for each 1-degree increase in Celsius. The intercept 32 is the Fahrenheit temperature when Celsius is 0.
03

Plot the Graph

To sketch the graph, pick two values for \( C \), calculate the corresponding \( F \), and plot them. For example, when \( C = 0 \), \( F = 32 \); and when \( C = 100 \), \( F = \frac{9}{5} \times 100 + 32 = 212 \). Plot the points \((0, 32)\) and \((100, 212)\) and draw a straight line through them.
04

Interpret the Graph

The line on the graph shows the direct proportionality between Celsius and Fahrenheit. The positive slope \( \frac{9}{5} \) indicates that as Celsius increases, Fahrenheit increases at a rate of 1.8 (or \( \frac{9}{5} \)). The point (0, 32) is the F-intercept, which means at 0掳C, it is 32掳F.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Temperature Conversion
Temperature conversion is an essential concept when dealing with different temperature scales such as Fahrenheit and Celsius. The formula to convert Celsius to Fahrenheit is given by the linear equation \( F = \frac{9}{5}C + 32 \). This equation makes it possible to determine the Fahrenheit equivalent of any Celsius temperature easily.

Here's how you can understand the conversion process:
  • The formula starts with multiplying the Celsius temperature \( C \) by \( \frac{9}{5} \). This part adjusts the scale, as each 1掳C increase is equivalent to a 1.8掳F increase.
  • Then, you add 32 to account for the offset between the Fahrenheit and Celsius scales. This is because 0掳C corresponds to 32掳F.
In practical applications, understanding this conversion is crucial in science, medicine, and weather forecasting, where both scales may be used.
Slope Interpretation
When dealing with linear functions, the slope is a fundamental concept to grasp. It represents the rate of change within the equation. For the temperature conversion formula \( F = \frac{9}{5} C + 32 \), the slope \( m \) is \( \frac{9}{5} \).

Interpreting the slope in the context of temperature:
  • It indicates that for every 1掳C increase, the Fahrenheit temperature increases by 1.8掳F. This illustrates a proportional relationship between the two scales.
  • The slope is positive, representing that as one variable increases, the other also increases.
Understanding the slope helps us see how changes in one variable affect the other, providing a clearer picture of real-world phenomena.
Graph Sketching
Graph sketching is a visual way to represent the relationship between two variables. In the case of the temperature conversion, we want to plot the Celsius temperature \( C \) versus the Fahrenheit temperature \( F \) based on the linear equation \( F = \frac{9}{5} C + 32 \).

To sketch the graph:
  • Identify key points that result from easy calculations, such as when \( C = 0 \) resulting in \( F = 32 \), and \( C = 100 \) resulting in \( F = 212 \). These represent significant points (like freezing and boiling in Celsius).
  • Plot these points on a coordinate plane. For example, point \((0, 32)\) shows 0掳C is equal to 32掳F.
  • Draw a straight line through these points. The graph is a straight line because it represents a linear function.
Graph sketching not only helps visualize the data but also aids in understanding the continuous rate of change between the two temperatures.

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Most popular questions from this chapter

Graph several members of the family of functions $$f(x)=\frac{1}{1+a e^{b x}}$$ where \(a>0 .\) How does the graph change when \(b\) changes? How does it change when \(a\) changes?

Let \(f(x)=\sqrt{1-x^{2}}, 0 \leqslant x \leqslant 1\) (a) Find \(f^{-1} .\) How is it related to \(f ?\) (b) Identify the graph of \(f\) and explain your answer to part (a).

Data points \((x, y)\) are given. (a) Draw a scatter plot of the data points. (b) Make semilog and log-log plots of the data. (c) Is a linear, power, or exponential function appropriate for modeling these data? (d) Find an appropriate model for the data and then graph the model together with a scatter plot of the data. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {2} & {4} & {6} & {8} & {10} & {12} \\\ \hline y & {0.08} & {0.12} & {0.18} & {0.26} & {0.35} & {0.53} \\\ \hline\end{array}$$ \(\begin{array}{|c|c|c|c|c|c|c|}\hline x & {1.0} & {2.4} & {3.1} & {3.6} & {4.3} & {4.9} \\ \hline y & {3.2} & {4.8} & {5.8} & {6.2} & {7.2} & {7.9} \\\ \hline\end{array}\)

When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by $$Q(t)=Q_{0}\left(1-e^{-t / a}\right)$$ (The maximum charge capacity is \(Q_{0}\) and \(t\) is measured inseconds.) (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90\(\%\) of capacity if \(a=2 ?\)

Find a formula for the inverse of the function. \(y=\frac{e^{x}}{1+2 e^{x}}\)
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