Question

We found a relationship between Fahrenheit temperature and Celsius temperature in Example \(8 .\) (a) Is there a temperature at which a Fahrenheit thermometer and a Celsius thermometer give the same reading? If so, what is it? (b) Writing to Learn Graph \(y_{1}=(9 / 5) x+32, y_{2}=\) \((5 / 9)(x-32),\) and \(y_{3}=x\) in the same viewing window. Explain how this figure is related to the question in part (a).

Short answer

The temperature at which Fahrenheit and Celsius thermometers give the same reading is -40 degrees. The graph confirms our solution because the intersection point of the three plotted functions corresponds to this temperature.

Step-by-step solution

- Solving for F = C

The relationship in the equations for \(F = (9/5)C + 32\) and \(C = (5/9)(F-32)\) shows that when they are equal, they satisfy the requirements of part (a) of the problem. So we set \(C = F\), and use either equation to solve for \(C\) and \(F\). Let's use: \(C = (5/9)(F-32)\). Setting \(F = C\) we get \(C = (5/9)(C-32)\). To solve for C, isolate C to one side by multiplying both sides by \(9/5\) and adding 32. This gives \(C = -40\). Hence, \(F = C = -40\) is the solution.

- Graphing the functions

Plotting the three equations on the graph will help us visualize their relationship. Plot \(y_{1} = (9 / 5)x + 32\) as the Fahrenheit function, \(y_{2} = (5 / 9)(x - 32)\) as the Celsius function, and \(y_{3} = x\) as the line where Fahrenheit equals Celsius. The intersection point of these three lines represents the common temperature point, which we found to be \(-40\) in our calculations in Step 1, thereby verifying the solution. As a result, it is evident the result from (a) corresponds with the plotted graph in (b).