Answer to Question #265123 in Algebra for jall

Question

A student drops a ball and measures the height the ball reaches after each bounce. She recorded the data in a table as shown.

Bounce numberMaximum Height after the bounce (cm) 1, 2,3,4 112,

89.6 ,71.68 ,57.32

1. From what height was the ball dropped? Show how you deduced your answer. (3 pts)
2. What percentage of the previous height will the ball reach on its next bounce? (2 pts)
3. On which bounce was the height reached less than 10 cm? (3 pts)
4. What was the height the ball reached after the 6th bounce? (2 pts)
5. Name a limitation to this mathematical model.

Answer 1

1.

$$h_0, h_1, h_2, h_3, h_4$$

$$\dfrac{h_1}{h_0}=\dfrac{h_2}{h_1}=\dfrac{h_3}{h_2}=\dfrac{h_4}{h_3}$$

$$\dfrac{112}{h_0}=\dfrac{89.6}{112}=\dfrac{71.68}{89.6}=\dfrac{57.32}{71.68}=0.8$$

$$h_0=\dfrac{112}{0.8}=140$$

The initial height is $140$ cm.

2.

$$\dfrac{h_1}{h_0}=\dfrac{h_2}{h_1}=\dfrac{h_3}{h_2}=\dfrac{h_4}{h_3}=0.8$$

The ball will reach 80 % of the previous height on its next bounce.

3.

$$140(0.8)^n<10$$

$$(0.8)^n<\dfrac{1}{14}$$

$$n>\dfrac{\ln(1/14)}{\ln(0.8)}, n\in \N$$

$$n\geq12$$

Check

$$140(0.8)^{11}=12.026>10$$

$$140(0.8)^{12}=9.621<10$$

The height was reached less than 10 cm at 12th bounce.

4.

$$140(0.8)^6=36.70016$$

The ball reaches the height of $36.70016$ cm after the 6th bounce.

5.

If there is no limit on number of bounces for the model , then it becomes a case of infinite Geometric Progression

$$S=h_0+\dfrac{2h_1}{1-0.8}=140+\dfrac{2(112)}{0.2}=1260(cm)$$

Total distance covered by the ball will never exceed 12.6m.