The question asks for the derivation of a linear function that relates temperatures in Celsius ($C$) and Fahrenheit ($F(C)$) using the form $F(C) = m \cdot C + t$. The given conditions are:

1. Water freezes at $0^\circ C$ and $32^\circ F$.
2. Water boils at $100^\circ C$ and $212^\circ F$.

Step 1: Solve for the slope ($m$)

The slope $m$ is determined by the formula for a straight line: $m = \frac{\Delta F}{\Delta C} = \frac{F_2 - F_1}{C_2 - C_1}.$

Substituting the points $(C_1, F_1) = (0, 32)$ and $(C_2, F_2) = (100, 212)$: $m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8.$

Step 2: Solve for the intercept ($t$)

Using the general equation $F(C) = m \cdot C + t$, substitute $m = 1.8$ and one of the points, such as $(0, 32)$: $32 = 1.8 \cdot 0 + t.$ $t = 32.$

Final Function:

The linear function is: $F(C) = 1.8 \cdot C + 32.$

Results:

• $m = 1.8$,
• $t = 32$.

Let me know if you want a deeper explanation or assistance with similar problems!

Questions for deeper understanding:

1. Why is the relationship between Celsius and Fahrenheit linear?
2. How would you rewrite this equation to convert Fahrenheit to Celsius?
3. What happens if the freezing and boiling points of water change under different conditions (e.g., altitude)?
4. Can you use this formula to calculate other points on the temperature scale, like $50^\circ C$?
5. What is the significance of the slope $m = 1.8$ in practical terms?

Tip: Always double-check units and linear relationships when working with temperature conversions!