Problem 120 - College Algebra Essentials

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 for Fahrenheit temperature corresponding to Celsius range from \( 15^{\circ}\) to \(35^{\circ}\) is \([59^{\circ}F, 95^{\circ}F]\)

Step 1: Identify conversion formula

First, identify the conversion formula which is given as \(C=\frac{5}{9}(F-32)\).

Step 2: Convert low end of Celsius range to Fahrenheit

Substitute the lower end of the Celsius range which is \(15^{\circ}C\) into the conversion formula: \[ F = \frac{9}{5}C + 32 = \frac{9}{5} \cdot 15 + 32 = 59^{\circ}F \].

Step 3: Convert high end of Celsius range to Fahrenheit

Substitute the higher end of the Celsius range which is \(35^{\circ}C\) into the conversion formula: \[ F = \frac{9}{5}C + 32 = \frac{9}{5} \cdot 35 + 32 = 95^{\circ}F \].

Step 4: Express the range in Interval Notation

Finally, express the range of Fahrenheit in interval notation. The interval notation range would be \([59, 95]\)