Ex 2.5, 3

Transcript

Ex 2.5, 3 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. (Hint: When °C = 0, °F = 32 and when °C = 100, °F = 212. Use this Information to find a and b, and thus, the linear relationship Between °C and °F.) Given our linear relationship °C = a °F + b Here, °C = Temperature in Celsius °F = Temperature in Fahrenheit We need to find values of a & b Given that ice melts at 0 degrees Celsius and 32 degrees Fahrenheit So, when °C = 0, °F = 32 Putting it in our equation 0 = a × 32 + b 0 = 32a + b 32a + b = 0 Also, water boils at 100 degrees Celsius and 212 degrees Fahrenheit. So, when °C = 100, °F = 212 Putting it in our equation 100 = a × 212 + b 100 = 212a + b 212a + b = 100 Now, our equations are 32a + b = 0 …(1) 212a + b = 100 …(2) Doing (2) – (1) (212a + b) – (32a + b) = 100 – 0 212a + b – 32a – b = 100 (212a – 32a) + (b – b) = 100 180a = 100 a = 100/180 a = 10/18 a = 𝟓/𝟗 Putting a = 20 in (1) 32a + b = 0 32 × 𝟓/𝟗 + b = 0 160/9 + b = 0 b = 0 – 160/9 b = (−𝟏𝟔𝟎)/𝟗 Now, putting values of a & b in our expression °C = a °F + b °C = 5/9 × °F + ((−160)/9) °C = 𝟓/𝟗 × °F – 𝟏𝟔𝟎/𝟗 Taking 5/9 common °C = 5/9 (°F – 160/9 ×9/5) °C = 𝟓/𝟗 (°F – 32)