Question

a. Find the linear function \(C=f(F)\) that gives the reading on the Celsius temperature scale corresponding to a reading on the Fahrenheit scale. Use the facts that \(C=0\) when \(F=32\) (freezing point) and \(C=100\) when \(F=212\) (boiling point). b. At what temperature are the Celsius and Fahrenheit readings equal?

Short answer

Answer: -40 degrees

Step-by-step solution

Find the Linear Function

Using the given information, we know that the Celsius scale has two points: \((32, 0)\) and \((212, 100)\). To find the linear function, we can use the slope-point formula. The formula for a linear function is given by \(C = m(F - F_1) + C_1\) where m is the slope and F is the Fahrenheit value.

Calculate the Slope

Using the given points, we can find the slope (m) using the formula: \(m=\frac{C_2-C_1}{F_2-F_1}\) Substitute the values from the given points \((F_1,C_1) = (32, 0)\) and \((F_2,C_2) = (212, 100)\): \(m=\frac{100-0}{212-32}= \frac{100}{180}=\frac{5}{9}\) Now we have the slope, m.

Calculate the Linear Function

Now that we have the slope, we can use the point-slope form of the linear equation to find the linear function C. We use the point \((32, 0)\) and the slope \(\frac{5}{9}\) to get: \(C = \frac{5}{9}(F - 32)\)

Find the Temperature

Now that we have the linear function \(C = \frac{5}{9}(F - 32)\), we must find the temperature at which \(C = F\). To do this, we will set Fahrenheit equal to Celsius and solve for F: \(\frac{5}{9}(F - 32) = F\) Now, multiply both sides by 9 to eliminate the fraction: \(5(F - 32) = 9F\) Expand the equation: \(5F - 160 = 9F\) Subtract 5F from both sides: \(-160 = 4F\) Finally, divide by 4 to obtain the Fahrenheit temperature: \(F = -40\) Therefore, the Celsius and Fahrenheit readings are equal at \(-40\) degrees.