Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month.
- (i) Find the height after 7 months.
- (ii) Make a table of values for t varying from 0 to 10 months and show how the height, h, increases every month.
- (iii) Find an expression that relates h and t, and explain why it represents linear growth.
Step-by-step solution
Idea: Start value 1.75 ft, fixed increase 0.5 ft per month. After t months the plant has grown 0.5t feet.
(i) Find the height after 7 months.
- Growth in 7 months = 0.5 × 7 = 3.5 feet.½ mark
- Height = 1.75 + 3.5 = 5.25 feet.½ mark
(ii) Make a table of values for t varying from 0 to 10 months and show how the height, h, increases every month.
- Add 0.5 ft for each month:1 mark
Month, t Height, h (feet) 0 1.75 1 2.25 2 2.75 3 3.25 4 3.75 5 4.25 6 4.75 7 5.25 8 5.75 9 6.25 10 6.75 - Each month the height increases by the same 0.5 ft.½ mark
(iii) Find an expression that relates h and t, and explain why it represents linear growth.
- h = 1.75 + 0.5t1 mark
- It is a linear (degree 1) expression in t with a positive coefficient 0.5: the height increases by a constant 0.5 ft over each equal interval of one month. So it represents linear growth.½ mark
Check: From the formula, h(7) = 1.75 + 3.5 = 5.25, the same as in (i) and in the table ✓.
Answer to write in the exam
(i)
Growth in 7 months = 0.5 × 7 = 3.5 feet
Height = 1.75 + 3.5
∴ Height after 7 months = 5.25 feet
(ii)
t: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
h: 1.75, 2.25, 2.75, 3.25, 3.75, 4.25, 4.75, 5.25, 5.75, 6.25, 6.75
∴ h increases by 0.5 feet every month
(iii)
h = 1.75 + 0.5t
h increases by a constant 0.5 feet in each month (equal intervals)
∴ It represents linear growth
Common mistakes that cost marks
- Writing h = 0.5 + 1.75t, swapping the starting height and the monthly growth.
- Starting the table at t = 0 with 2.25 ft. At t = 0 the plant has not grown yet: 1.75 ft.
- Explaining linear growth only as “it grows”. Say that it grows by the same amount in each equal time interval.
How this can come in the exam
A tree is 2.5 m tall and grows 0.4 m every year. Its height after 6 years is
- 4.9 m
- 2.9 m
- 5.4 m
- 3.9 m
Show answer
(A) 4.9 m
2.5 + 0.4 × 6 = 2.5 + 2.4 = 4.9 m.
A bamboo shoot is 12 cm tall when planted and grows 4.5 cm every day.
(i) Write its height h after d days. (ii) Find the height after 8 days. (iii) After how many days will it be 66 cm tall? (iv) Explain why this is linear growth.
Show answer
(i) h = 12 + 4.5d (1 mark). (ii) 12 + 36 = 48 cm (1 mark). (iii) 4.5d = 54 ⇒ d = 12 days (1 mark). (iv) It increases by the same 4.5 cm each day (1 mark).Try one yourself
A plant is 0.8 m tall and grows 0.15 m each month. Write h in terms of t and find h after 10 months.
Show answer
h = 0.8 + 0.15t; after 10 months 2.3 m.
More questions like this
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- The initial population of a village is 750. Every year, 50 people move from a nearby city to the village.
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- Thus, y = 20x + 150 represents the linear relationship between y, the bill amount in Rupees, and x, the number of GB of the internet data used.
Can you guess what the numbers 20 and 150 in the equation y = 20x + 150 represent?