Predicting What Comes Next: Sequences and Progressions | Exercise 8.3

Question 5

A ball is dropped from a height of 80 metres. After hitting the ground, it bounces back to 60% of the height from which it fell. It continues bouncing in this way—each time rising to 60% of the previous height.

(i) What height does the ball reach after the $5^{\text{th}}$ bounce? (ii) What is the total vertical distance the ball has travelled by the time it hits the ground for the $6^{\text{th}}$ time?

Solution

Understand the Question

The ball is initially dropped from a height of $80\text{ m}$ .
After each bounce, it reaches $60\% = 0.6$ of its previous height.
For part (i), calculate the height after each bounce up to the $5^{\text{th}}$ bounce.
For part (ii), by the time the ball hits the ground for the $6^{\text{th}}$ time, it has completed the initial downward drop plus $5$ complete bounces (each bounce involves travelling up and then back down, so distance $= 2 \times \text{height}$ ).

(i) What height does the ball reach after the $5^{\text{th}}$ bounce?

Step 1 · Calculate Height After Each Bounce

Initial height $= 80\text{ m}$ , bounce ratio $= 60\% = 0.6$ . Diagram 1

Height after 1st bounce:

\begin{aligned} &= 80 \times 0.6 \\ &= 48 \text{ m} \end{aligned}

Height after 2nd bounce:

\begin{aligned} &= 48 \times 0.6 \\ &= 28.8 \text{ m} \end{aligned}

Height after 3rd bounce:

\begin{aligned} &= 28.8 \times 0.6 \\ &= 17.28 \text{ m} \end{aligned}

Height after 4th bounce:

\begin{aligned} &= 17.28 \times 0.6 \\ &= 10.368 \text{ m} \end{aligned}

Height after 5th bounce:

\begin{aligned} &= 10.368 \times 0.6 \\ &= 6.2208 \text{ m} \end{aligned}

Answer

(i) $6.2208\text{ m}$

(ii) What is the total vertical distance the ball has travelled by the time it hits the ground for the $6^{\text{th}}$ time?

Step 1 · Calculate Total Vertical Distance

Distance of initial fall: $= 80 \text{ m}$

Distance for 5 bounces (rising and falling):

\begin{aligned} &= 2 \times (\text{Height after 1st bounce} + \text{Height after 2nd bounce} + \text{Height after 3rd bounce} + \text{Height after 4th bounce} + \text{Height after 5th bounce}) \\ &= 2 \times (48 + 28.8 + 17.28 + 10.368 + 6.2208) \\ &= 2 \times (110.6688) \\ &= 221.3376 \text{ m} \end{aligned}

Total vertical distance:

\begin{aligned} &= 80 + 221.3376 \\ &= 301.3376 \text{ m} \end{aligned}

Answer

(ii) $301.3376\text{ m}$

Common Mistakes

Forgetting Both Up and Down Paths: Each bounce covers the peak height twice (once going up and once coming down). Multiplying the bounce heights by $2$ is necessary.
Counting the Wrong Number of Bounces: The ball hits the ground for the $6^{\text{th}}$ time after the initial drop ( $1^{\text{st}}$ hit) and $5$ bounces. Do not include a $6^{\text{th}}$ bounce.

More questions in Exercise 8.3

Find the $12^{\text{th}}$ term of a GP with common ratio 2, whose $8^{\text{th}}$ term is 192.

Find the $10^{\text{th}}$ and $n^{\text{th}}$ terms of the GP: $5, 25, 125, \dots$ .

A sequence is given by the recursive rule $t_1 = 2$ , $t_{n+1} = 3t_n - 2$ for $n \ge 1$ . Which term of the sequence is 730?

Which term of the GP: $2, 6, 18, \dots$ is $4374$ ? Write the explicit formula as well as the recursive formula for the $n^{\text{th}}$ term.

(i) What height does the ball reach after the $5^{\text{th}}$ bounce? (ii) What is the total vertical distance the ball has travelled by the time it hits the ground for the $6^{\text{th}}$ time?

Which term of the sequence $2, 2\sqrt{2}, 4, \dots$ is 128?

Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. Stage 0 of this fractal is a square sheet of paper. To construct Stage 1, each side of the square is trisected and the points of trisection of opposite sides are joined to obtain nine smaller squares. The centre square is then removed and the 8 smaller squares are retained, leaving a square hole in the centre. The same process is repeated on the eight smaller shaded squares to obtain Stage 2 and so on.

Look at Fig. 8.12 and try to answer the following questions.

(i) How many red squares are there in Stages 0 to 3? (ii) Can you predict the number of red squares in Stages 4 and 5? (iii) Can you find a rule for the number of red squares at the $n^{\text{th}}$ stage? Write the explicit formula as well as the recursive formula for the number of red squares at any stage. (iv) Suppose the area of the square in Stage 0 is 1 square unit. What is the area of the red region in Stages 1, 2 and 3? What will be the area of the red region in Stages 4 and 5? Find the explicit as well as the recursive formula for the area of the red region at the $n^{\text{th}}$ stage. What happens to this area as $n$ , the number of stages, goes on increasing?