A ball is dropped from a height of 24 feet above the ground. Each time the ball bounces up to (34)th of its previous height.
- (a) Can you write the sequence of numbers obtained from the heights attained by the ball in five successive bounces?
- (b) How many bounces are required for the ball to remain below a height of 16 of the original height from which it was dropped?
Step-by-step solution
Idea: Each bounce height is 34 = 0.75 of the one before, so the heights form a GP. Keep multiplying by 0.75 until the height drops below 4 feet.
(a) Can you write the sequence of numbers obtained from the heights attained by the ball in five successive bounces?
- 1st bounce: 24 × 0.75 = 18 ft; 2nd: 18 × 0.75 = 13.5 ft; 3rd: 13.5 × 0.75 = 10.125 ft.1 mark
- 4th: 10.125 × 0.75 = 7.59375 ft; 5th: 7.59375 × 0.75 = 5.6953125 ft (≈ 5.695 ft).½ mark
- Sequence: 18, 13.5, 10.125, 7.59375, 5.695, … a GP with a = 18 and r = 34.½ mark
(b) How many bounces are required for the ball to remain below a height of 16 of the original height from which it was dropped?
- 16 of the original height = 16 × 24 = 4 feet.½ mark
- 6th bounce: 5.6953 × 0.75 ≈ 4.27 ft (still above 4). 7th bounce: 4.27 × 0.75 ≈ 3.20 ft (below 4).1 mark
- Every later bounce is lower still, so after the 7th bounce the ball stays below 16 of its starting height.½ mark
Check: Using the nth term 18 × (0.75)ⁿ⁻¹: n = 6 gives 4.27 > 4 and n = 7 gives 3.20 < 4 ✓.
Answer to write in the exam
(a)
1st bounce = 24 × 0.75 = 18 ft
2nd = 18 × 0.75 = 13.5 ft
3rd = 13.5 × 0.75 = 10.125 ft
4th = 10.125 × 0.75 = 7.59375 ft
5th = 7.59375 × 0.75 ≈ 5.695 ft
∴ 18, 13.5, 10.125, 7.59375, 5.695 (GP, a = 18, r = 34)
(b)
16 × 24 = 4 ft
6th bounce = 5.695 × 0.75 ≈ 4.27 ft > 4 ft
7th bounce = 4.27 × 0.75 ≈ 3.20 ft < 4 ft
∴ 7 bounces are required.
Common mistakes that cost marks
- Taking 24 as the first bounce height. 24 ft is the drop height; the first bounce reaches 18 ft.
- Comparing with 16 of the previous height instead of 16 of the original 24 ft (= 4 ft).
- Answering 6 because 4.27 is ‘about 4’. It must be below 4 ft, which first happens at the 7th bounce.
How this can come in the exam
A ball dropped from 32 m rises to 12 of its previous height each bounce. The height after the 4th bounce is
- 4 m
- 2 m
- 1 m
- 8 m
Show answer
(B) 2 m
16, 8, 4, 2.
A ball is dropped from 81 cm and each bounce reaches 23 of the previous height. After how many bounces is the height first below 20 cm?
Show answer
Heights: 54, 36, 24, 16 cm (2 marks). 16 < 20 first at the 4th bounce (1 mark).Try one yourself
A ball dropped from 50 m rises to 45 of its previous height after each bounce. Write the heights of the first four bounces.
Show answer
40, 32, 25.6, 20.48 m (a GP with r = 45).
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