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Linear relationships · 3 marks

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.)

Answer: a = 59 and b = −1609, so °C = 59 °F − 1609, that is, °C = 59(°F − 32).

Step-by-step solution

Given: °C = 0 when °F = 32; °C = 100 when °F = 212; °C = a °F + b
To find: a and b

Idea: Here °C plays the part of y and °F the part of x. Put each pair of values into °C = a °F + b to get two equations.

  1. Ice: 0 = 32a + b … (1). Boiling water: 100 = 212a + b … (2).1 mark
  2. From (1), b = −32a. In (2): 100 = 212a − 32a = 180a ⇒ a = 100180 = 59.1 mark
  3. b = −32 × 59 = −1609 (about −17.8).½ mark
  4. So °C = 59 °F − 1609 = 59(°F − 32).½ mark
a = 59 and b = −1609, so °C = 59 °F − 1609 = 59(°F − 32).

Check: °F = 32: 59 × 0 = 0 °C ✓. °F = 212: 59 × 180 = 100 °C ✓. Body temperature 98.6 °F: 59 × 66.6 = 37 °C, as expected.

Answer to write in the exam

0 = 32a + b … (1); 100 = 212a + b … (2)

From (1): b = −32a

In (2): 100 = 212a − 32a = 180a ⇒ a = 59

b = −32 × 59 = −1609

∴ a = 59, b = −1609; °C = 59(°F − 32)

Common mistakes that cost marks

  • Putting the values the wrong way round, for example 32 = 0a + b. In °C = a °F + b, the Fahrenheit value multiplies a.
  • Writing a = 95. That is the rate for changing °C to °F; here °C is written in terms of °F, so a = 59.
  • Taking b = 32 or b = −32. From (1), b = −32a = −1609.

How this can come in the exam

MCQ (1 mark)

Using °C = 59(°F − 32), the temperature 50 °F equals

  1. 10 °C
  2. 18 °C
  3. 45 °C
  4. −10 °C
Show answer

(A) 10 °C
59 × 18 = 10 °C.

Assertion–Reason (1 mark)

Assertion (A): −40 °F and −40 °C are the same temperature.
Reason (R): 59(−40 − 32) = −40.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
59 × (−72) = −40, so −40 °F is −40 °C. R is the calculation that proves A.

Try one yourself

Using the relation found above, convert 113 °F to °C.

Show answer

59(113 − 32) = 59 × 81 = 45 °C.

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