Consider the relationship between temperature measured in degrees Celsius (°C) and degrees Fahrenheit (°F), which is given by °C = a°F + b. Find a and b, given that ice melts at 0 degrees Celsius and 32 degrees Fahrenheit, and water boils at 100 degrees Celsius and 212 degrees Fahrenheit.
a = 5/9, b = -160/9
Explanation
The problem requires finding the constants 'a' and 'b' in the linear relationship °C = a°F + b. Using the given reference points for the melting point of ice (0°C = 32°F) and the boiling point of water (100°C = 212°F), we can set up a system of two linear equations. Solving this system by elimination gives the values of a and b, which establishes the linear relationship between Celsius and Fahrenheit scales.
Solution Steps
Step 1: Formulate equations using the given conditions. When °C = 0, °F = 32. Substituting in °C = a°F + b, we get: 0 = 32a + b ...(i)
Step 2: When °C = 100, °F = 212. Substituting in °C = a°F + b, we get: 100 = 212a + b ...(ii)
Step 3: Solve the equations to find 'a'. Subtract equation (i) from equation (ii): (212a + b) - (32a + b) = 100 - 0 180a = 100 a = 100/180 a = 5/9
Step 4: Solve for 'b' by substituting the value of 'a' into equation (i): 0 = 32(5/9) + b b = -160/9