Question

Temperature Conversion \(\quad\) Find a linear equation that expresses the relationship between the temperature in degrees Celsius \(C\) and the temperature in degrees Fahrenheit \(F\). Use the fact that water freezes at \(0^{\circ} \mathrm{C}\left(32^{\circ} \mathrm{F}\right)\) and boils at \(100^{\circ} \mathrm{C}\left(212^{\circ} \mathrm{F}\right)\). Use the equation to convert \(72^{\circ} \mathrm{F}\) to degrees Celsius.

Short answer

The linear equation relating Celsius and Fahrenheit is \( F = \frac{5}{9}C + 32 \). Using this equation, we find that 72°F is equivalent to 40°C.

Step-by-step solution

Calculating the Slope

To find the slope 'm' of the line, we use the formula \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \). In this case, our two points are (32, 0) and (212, 100). So, \( m=\frac{100-0}{212-32} = \frac{100}{180} = \frac{5}{9} \). Therefore, the slope of the line is \(\frac{5}{9}\).

Finding the y-Intercept

The y-intercept 'c' is the temperature in Fahrenheit when Celsius is 0. It is given that this value is 32. So, 'c' equals 32.

Formulating the Linear Equation

Our linear equation formulated from the slope and y-intercept values is \( F=\frac{5}{9}C + 32\).

Conversion

To convert 72°F to Celsius, we rewrite the equation in terms of C and insert 72 as the value of F. We get, \( C = \frac{9}{5}(F-32) \) and inserting F=72, we get \( C = \frac{9}{5}(72-32) \) which simplifies to 40 degrees Celsius.