When a tennis ball is dropped, it rebounds to 70% of its initial height. If we drop a tennis ball from 10 ft, determine the total distance traveled by the ball after 10 bounces.
70% = 7 ft (10ft x 70% = 7ft)
after 10 bounces
10 x 7ft = 70 ft
To determine the total distance traveled by the tennis ball after 10 bounces, we need to calculate the sum of all the distances traveled during each bounce.
First, let's calculate the distance traveled by the ball on the first bounce. Since the ball rebounds to 70% (0.7) of its initial height, it reaches a height of 10 ft * 0.7 = 7 ft on the first bounce. The total distance covered during the first bounce is twice the height, so it is 2 * 7 ft = 14 ft.
For the subsequent bounces, we need to find the distance traveled during each bounce and add it to the total distance.
On the second bounce, the ball reaches a height of 7 ft * 0.7 = 4.9 ft. The total distance covered during the second bounce is 2 * 4.9 ft = 9.8 ft.
On the third bounce, the ball reaches a height of 4.9 ft * 0.7 = 3.43 ft. The total distance covered during the third bounce is 2 * 3.43 ft = 6.86 ft.
We can continue this process for all 10 bounces and sum up the distances. However, to simplify the calculation, we can notice that the distances are decreasing geometrically.
The total distance for all 10 bounces can be calculated using the formula for the sum of a geometric series:
Sn = a(1 - r^n) / (1 - r),
where Sn is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 14 ft, r = 0.7, and n = 10.
Plugging these values into the formula, we get:
Sn = 14(1 - 0.7^10) / (1 - 0.7).
Calculating this expression will give us the total distance traveled by the ball after 10 bounces.