In countries like the USA, temperature is measured in Fahrenheit, whereas in countries like India, it is measured in Celsius. Here is a linear equation that converts Celsius to Fahrenheit:
F = 95C + 32
- (i) Draw the graph of the linear equation above using Celsius on the x-axis and Fahrenheit on the y-axis.
- (ii) If the temperature is 30 °C, what is the temperature in Fahrenheit?
- (iii) If the temperature is 95 °F, what is the temperature in Celsius?
- (iv) If the temperature is 0 °C, what is the temperature in Fahrenheit and if the temperature is 0 °F, what is the temperature in Celsius?
- (v) Is there a temperature which is numerically the same in both Fahrenheit and Celsius? If yes, find it.
Answer: (i) A straight line through (0, 32) and (−40, −40), slope 95 (ii) 86 °F (iii) 35 °C (iv) 0 °C = 32 °F; 0 °F = −1609 ≈ −17.8 °C (v) Yes, −40 (−40 °C = −40 °F).
Step-by-step solution
Idea: This is a linear equation in two variables C and F. To convert, substitute the known value and solve for the other. The graph is a line with slope 95 and F-intercept 32.
(i) Draw the graph of the linear equation above using Celsius on the x-axis and Fahrenheit on the y-axis.
- Table: C = 0 → F = 32; C = 30 → F = 86; C = −40 → F = −40.
Plot these points and join them with a straight line (see the graph).1 markC −40 0 30 F −40 32 86
A straight line through (−40, −40), (0, 32) and (30, 86)
(ii) If the temperature is 30 °C, what is the temperature in Fahrenheit?
- F = 95 × 30 + 32 = 54 + 32 = 86 °F.1 mark
86 °F
(iii) If the temperature is 95 °F, what is the temperature in Celsius?
- 95 = 95C + 32 ⇒ 95C = 63 ⇒ C = 63 × 59 = 35 °C.1 mark
35 °C
(iv) If the temperature is 0 °C, what is the temperature in Fahrenheit and if the temperature is 0 °F, what is the temperature in Celsius?
- C = 0: F = 0 + 32 = 32 °F. F = 0: 95C = −32 ⇒ C = −1609 ≈ −17.8 °C. (These are where the line meets the two axes.)1 mark
32 °F; −1609 °C ≈ −17.8 °C
(v) Is there a temperature which is numerically the same in both Fahrenheit and Celsius? If yes, find it.
- Put F = C: C = 95C + 32 ⇒ −45C = 32 ⇒ C = −40. Yes: −40 °C = −40 °F (where the graph meets the line F = C).1 mark
Yes, −40
(i) Line through (−40, −40), (0, 32), (30, 86) (ii) 86 °F (iii) 35 °C (iv) 32 °F; −160/9 °C ≈ −17.8 °C (v) Yes, −40°.
Check: (iii) 95 × 35 + 32 = 63 + 32 = 95 ✓. (v) 95 × (−40) + 32 = −72 + 32 = −40 ✓.
Answer to write in the exam
(i)
| C | −40 | 0 | 30 |
| F | −40 | 32 | 86 |
Plot (−40, −40), (0, 32), (30, 86) and join: graph of F = 95C + 32
(ii)
F = 95 × 30 + 32 = 54 + 32
∴ F = 86 °F
(iii)
95 = 95C + 32 ⇒ 95C = 63
C = 63 × 59
∴ C = 35 °C
(iv)
C = 0 ⇒ F = 32 °F
F = 0 ⇒ 95C = −32 ⇒ C = −1609 ≈ −17.8 °C
(v)
Let F = C: C = 95C + 32
−45C = 32 ⇒ C = −40
∴ −40 °C = −40 °F
Common mistakes that cost marks
- In (iii), subtracting 32 after dividing instead of before: first 95 − 32 = 63, then × 59.
- Thinking 0 °F is the same as 0 °C.
- Using equal scales on both axes and running out of paper; a different scale on each axis is fine if it is marked.
How this can come in the exam
MCQ (1 mark)
Using F = 95C + 32, 100 °C equals
- 180 °F
- 212 °F
- 132 °F
- 100 °F
Show answer
(B) 212 °F
180 + 32 = 212.
Short answer (2 marks)
Convert 59 °F to Celsius using F = 95C + 32.
Show answer
59 − 32 = 27 = 95C (1 mark) ⇒ C = 15 °C (1 mark).Try one yourself
Convert 20 °C to °F and 50 °F to °C.
Show answer
95 × 20 + 32 = 68 °F; (50 − 32) × 59 = 10 °C.
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