The height of water in a cylindrical tank is 3 m at the start of summer. The height h m at the end of t months is given by the linear function h(t) = 3 − 0.5t.
Step-by-step solution
Idea: Substitute t = 0, 1, 2, … The coefficient of t is −0.5, so each month the height goes down by 0.5 m.
- 1 mark
Month, t 0 1 2 3 4 Height, h (m) 3 2.5 2 1.5 1 - As t increases by a fixed amount (one month), h decreases by a fixed amount (0.5 m).½ mark
- A quantity that decreases by a constant amount over equal intervals shows linear decay.½ mark
Check: h(4) = 3 − 2 = 1 ✓.
Answer to write in the exam
h(t) = 3 − 0.5t
t = 0, 1, 2, 3, 4 → h = 3, 2.5, 2, 1.5, 1 m
Each month h decreases by 0.5 m
∴ The height shows linear decay
Common mistakes that cost marks
- Writing h(1) = 3.5, adding 0.5 instead of subtracting it.
- Working out 0.5t for t = 3 as 0.15 instead of 1.5.
- Calling this linear growth because t is increasing. What matters is the quantity h, which is decreasing.
How this can come in the exam
Which function shows linear decay?
- f(t) = 5 + 2t
- f(t) = 40 − 3t
- f(t) = t2
- f(t) = 7t
Show answer
(B) f(t) = 40 − 3t
40 − 3t falls by the same 3 for each unit increase in t.
Assertion (A): h(t) = 3 − 0.5t represents linear decay.
Reason (R): In h(t), the coefficient of t is negative, so h drops by the same amount each month.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
Both are true, and the negative constant rate of change (R) is exactly why it is linear decay (A).
Try one yourself
The fuel in a generator is F(t) = 20 − 2.5t litres after t hours. Make a table for t = 0 to 4.
Show answer
20, 17.5, 15, 12.5, 10 litres (down 2.5 litres every hour: linear decay).
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- The height of water in a cylindrical tank is 3 m at the start of summer. The height h m at the end of t months is given by the linear function h(t) = 3 − 0.5t.
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