Question

[maximum mark: 7]22m.1.sl.tz1.13 a ball is dropped from a height of metres and bounces on the ground. the maximum height reached by the ball, after each bounce, is of the previous maximum height. (a) show that the maximum height reached by the ball after it has bounced for the sixth time is , to the nearest .[2] (b) find the number of times, after the rst bounce, that the maximum height reached is greater than .[2] (c) find the total vertical distance travelled by the ball from the point at which it is dropped until the fourth bounce.[3] 1.8 85% 68cm cm 10cm

Answer 1

(a) The maximum height reached by the ball after it has bounced for the sixth time is approximately 0.89 meters.

(b) The number of times, after the first bounce, that the maximum height reached is greater than 1.8 meters is 3 times.

(c) The total vertical distance traveled by the ball from the point at which it is dropped until the fourth bounce is approximately 6.47 meters.

Step-by-step explanation:

(a) When the ball bounces, it reaches a maximum height that is a fraction of the previous maximum height. The general formula for the maximum height after bouncing n times is
`\( h_n = h_0 * \left((1)/(2)\right)^n \)`, where h₀ is the initial height. Substituting n = 6 and h₀ = 1.8 meters, we get h₆ ≈ 0.89 meters.

(b) To find the number of times the maximum height exceeds 1.8 meters, we need to solve the inequality
`\( h_n > 1.8 \)`. Setting
`\( h_0 * \left((1)/(2)\right)^n > 1.8 \)`and solving for n , we find that n > 2 . Therefore, the maximum height exceeds 1.8 meters after 3 bounces.

(c) The total vertical distance traveled by the ball until the fourth bounce is the sum of the distances traveled during each bounce. Using the formula for the sum of a geometric series,
`\( S_n = h_0 * \left((1-\left((1)/(2)\right)^n)/(1-(1)/(2))\right) \)`, with n = 4 and h₀ = 1.8, we find S₄ ≈ 6.47 meters.

Answer 2

Using the formula for the height after each bounce, we can calculate the specific heights after the sixth bounce, determine the number of bounces over 10cm height, and calculate the total vertical distance traveled by the ball until the fourth bounce.

Step-by-step explanation:

Calculation of Bounce Heights and Total Distance Traveled

To address the initial student's question which appears to be incomplete, let's assume a ball is dropped from a certain height and after each bounce, it reaches 85% of its previous maximum height. We can use this information to calculate the following:

1. Show the maximum height reached by the ball after the sixth bounce.
2. Find the number of times the ball reaches a height greater than 10cm after the first bounce.
3. Calculate the total vertical distance the ball travels until the fourth bounce.

For part (a), assuming the ball is dropped from 1.8 meters, we can calculate the maximum height after the sixth bounce using the formula: height after nth bounce = initial height * (0.85)^n. So after the sixth bounce, the height will be 1.8m * (0.85)^6, which we need to calculate and round to the nearest cm.

For part (b), we need to determine how many times the ball, when dropped, will exceed 10cm in height after bouncing. This will involve determining the exact bounce at which the ball's height falls below 10cm.

For part (c), the vertical distance traveled by the ball until the fourth bounce includes both the descent and ascent for each bounce. We sum the initial drop and the heights of subsequent bounces:

• Descent: 1.8m
• First ascent and second descent: 1.8m * 85% * 2
• Second ascent and third descent: 1.8m * (85%)^2 * 2
• Third ascent and fourth descent: 1.8m * (85%)^3 * 2

Adding these distances will give the total vertical distance traveled until the fourth bounce.