Problem 120 - College Algebra

Question

The formula for converting Fahrenheit temperature, \(F,\) to Celsius temperature, \(C\), is $$C=\frac{5}{9}(F-32)$$ If Celsius temperature ranges from \(15^{\circ}\) to \(35^{\circ},\) inclusive, what is the range for the Fahrenheit temperature? Use interval notation to express this range.

Step-by-Step Solution

Answer

The range of the Fahrenheit temperature when the Celsius temperature is from 15 to 35 inclusive is [59, 95].

Step 1: Understanding the Formula

The formula for converting Fahrenheit to Celsius is \(C=\frac{5}{9}(F-32)\). This formula takes into account the different scalings and baselines of both the temperature systems. We can transpose this formula to convert Celsius to Fahrenheit as: \(F=\frac{9}{5}C+32\).

Step 2: Convert Lower Limit

Now we apply the formula to the lower limit of the Celsius temperature range, which is 15. Substituting \(C=15\) in the formula, we get: \(F=\frac{9}{5}*15+32=59\).

Step 3: Convert Upper Limit

Next, we apply the formula to the upper limit of the Celsius temperature range, which is 35. Substituting \(C=35\) into the formula, we get: \(F=\frac{9}{5}*35+32=95\).

Step 4: Determine Fahrenheit Range in Interval Notation

The range of Fahrenheit would thus be equal to the output of Fahrenheit for the Celsius range. Therefore, that range would be from 59 to 95 inclusive. In interval notation, this would be [59, 95].