Let's solve this step by step.

Part a) Finding the Linear Function F(C)

To find the linear function that converts degrees Celsius (C) to degrees Fahrenheit (F), we use the fact that the relationship between Celsius and Fahrenheit is linear. A linear function has the form:

$F(C) = mC + b$

Where:

• $F(C)$ is the temperature in Fahrenheit when the temperature is $C$ in Celsius,
• $m$ is the slope (rate of change),
• $b$ is the y-intercept (the value of $F$ when $C = 0$).

We are given two key points:

• At freezing point: $C = 0$, $F = 32$ (this gives the intercept $b = 32$),
• At boiling point: $C = 100$, $F = 212$.

Now, we find the slope $m$:

$m = \frac{\Delta F}{\Delta C} = \frac{F(100) - F(0)}{100 - 0} = \frac{212 - 32}{100} = \frac{180}{100} = 1.8$

So, the linear function is:

$F(C) = 1.8C + 32$

Part b) Converting 20ºC to Fahrenheit

Now that we have the function $F(C) = 1.8C + 32$, let's use it to convert 20ºC to Fahrenheit.

$F(20) = 1.8 \times 20 + 32 = 36 + 32 = 68$

So, $20^\circ C = 68^\circ F$.

Summary:

• Linear function: $F(C) = 1.8C + 32$
• Conversion of 20ºC to Fahrenheit: $20^\circ C = 68^\circ F$

Do you want more details or have any questions about this?

Related Questions:

1. How would you convert 50ºC to Fahrenheit using this function?
2. How would you derive the inverse function to convert Fahrenheit to Celsius?
3. What would be the Fahrenheit equivalent of -10ºC?
4. What is the Celsius temperature when Fahrenheit is 86ºF?
5. How does the slope $m$ of 1.8 relate to the rate of temperature conversion?

Tip: To quickly convert between Celsius and Fahrenheit, remember that for every 1ºC increase, Fahrenheit increases by 1.8º!