Answer :
To determine the total distance traveled by the ball when it is dropped from a height of 10 feet and stops after the 30th bounce, we can break down the process step-by-step.
### Step-by-step Analysis
1. Initial Drop:
- The ball drops from a height of 10 feet initially.
- Let [tex]\( x \)[/tex] be the height, so [tex]\( x = 10 \)[/tex].
Initial drop distance [tex]\( D_1 = x = 10 \)[/tex] feet.
2. Subsequent Bounces:
- Each time the ball hits the ground, it bounces back to [tex]\( \frac{2}{3} \)[/tex] of its previous height.
- The ball eventually stops at the 30th bounce.
For each bounce, let's define:
- [tex]\( h_1 \)[/tex] as the height after the first bounce:
[tex]\( h_1 = 10 \times \frac{2}{3} \)[/tex]
- [tex]\( h_2 \)[/tex] as the height after the second bounce:
[tex]\( h_2 = 10 \times \left(\frac{2}{3}\right)^2 \)[/tex]
- and so on.
### Calculation Summary
1. The distance for the first drop from 10 feet is already considered.
Total initial distance: [tex]\( D_{\text{initial}} = 10 \)[/tex] feet.
2. For the bounces, both upwards and downwards travels need to be considered:
- The ball goes up and then comes down for each bounce (except the first drop and last ascent).
#### Series Calculation:
For each bounce [tex]\( i \)[/tex] (where [tex]\( i \)[/tex] ranges from 1 to 30):
- Height reached after bounce [tex]\( i \)[/tex] up: [tex]\( h_i = 10 \times \left(\frac{2}{3}\right)^i \)[/tex]
- Thus for each bounce (for every up and down), the distance:
[tex]\( D_{\text{bounce}_i} = 2 \times 10 \times \left(\frac{2}{3}\right)^i \)[/tex]
The total distance [tex]\( D_{\text{total}} \)[/tex] traveled by the ball after 30 bounces will be the initial drop distance plus the sum of all bounce travels (both up and down).
Total distance covered:
[tex]\[ D_{\text{total}} = 10 + 2 \times \sum_{i=1}^{30} 10 \left(\frac{2}{3}\right)^i \][/tex]
#### Calculating the Series
We can recognize the sum as a geometric series:
[tex]\[ \sum_{i=1}^{30} \left(\frac{2}{3}\right)^i \][/tex]
This is simplified by multiplying by 10 and by 2.
Calculate the sum of the series where:
[tex]\[ S = \sum_{i=1}^{30} \left( \frac{2}{3} \right)^i \][/tex]
The geometric series sum formula for [tex]\( n \)[/tex] terms is:
[tex]\[ S_n = a \frac{1 - r^n}{1 - r} \][/tex]
where [tex]\( a = \frac{2}{3} \)[/tex], [tex]\( r = \frac{2}{3} \)[/tex], and [tex]\( n = 30 \)[/tex].
So, computing the series sum:
[tex]\[ S_{30} = \frac{2/3 \left(1 - \left(\frac{2}{3}\right)^{30}\right)}{1 - 2/3} \][/tex]
[tex]\[ S_{30} = 2 \left( \frac{1 - \left(\frac{2}{3}\right)^{30}}{1/3} \right) \][/tex]
[tex]\[ S_{30} = 6 \left( 1 - \left(\frac{2}{3}\right)^{30} \right) \][/tex]
The total distance for the bounces computation becomes:
[tex]\[ \text{Total Bounce Distance} = 2 \times 10 \times S_{30} = 20 \times 6 \left( 1 - \left(\frac{2}{3} \right)^{30} \right) \][/tex]
So, the total distance ball traveled:
[tex]\[ \approx 49.99979139619796 \text{ feet} \][/tex]
Thus, the total distance traveled by the ball is:
[tex]\[ 50 \, \text{feet} \][/tex] (rounded to the nearest integer).
So the total distance is approximately [tex]\( 5 \times 10^1 \)[/tex] feet in exponential notation.
### Step-by-step Analysis
1. Initial Drop:
- The ball drops from a height of 10 feet initially.
- Let [tex]\( x \)[/tex] be the height, so [tex]\( x = 10 \)[/tex].
Initial drop distance [tex]\( D_1 = x = 10 \)[/tex] feet.
2. Subsequent Bounces:
- Each time the ball hits the ground, it bounces back to [tex]\( \frac{2}{3} \)[/tex] of its previous height.
- The ball eventually stops at the 30th bounce.
For each bounce, let's define:
- [tex]\( h_1 \)[/tex] as the height after the first bounce:
[tex]\( h_1 = 10 \times \frac{2}{3} \)[/tex]
- [tex]\( h_2 \)[/tex] as the height after the second bounce:
[tex]\( h_2 = 10 \times \left(\frac{2}{3}\right)^2 \)[/tex]
- and so on.
### Calculation Summary
1. The distance for the first drop from 10 feet is already considered.
Total initial distance: [tex]\( D_{\text{initial}} = 10 \)[/tex] feet.
2. For the bounces, both upwards and downwards travels need to be considered:
- The ball goes up and then comes down for each bounce (except the first drop and last ascent).
#### Series Calculation:
For each bounce [tex]\( i \)[/tex] (where [tex]\( i \)[/tex] ranges from 1 to 30):
- Height reached after bounce [tex]\( i \)[/tex] up: [tex]\( h_i = 10 \times \left(\frac{2}{3}\right)^i \)[/tex]
- Thus for each bounce (for every up and down), the distance:
[tex]\( D_{\text{bounce}_i} = 2 \times 10 \times \left(\frac{2}{3}\right)^i \)[/tex]
The total distance [tex]\( D_{\text{total}} \)[/tex] traveled by the ball after 30 bounces will be the initial drop distance plus the sum of all bounce travels (both up and down).
Total distance covered:
[tex]\[ D_{\text{total}} = 10 + 2 \times \sum_{i=1}^{30} 10 \left(\frac{2}{3}\right)^i \][/tex]
#### Calculating the Series
We can recognize the sum as a geometric series:
[tex]\[ \sum_{i=1}^{30} \left(\frac{2}{3}\right)^i \][/tex]
This is simplified by multiplying by 10 and by 2.
Calculate the sum of the series where:
[tex]\[ S = \sum_{i=1}^{30} \left( \frac{2}{3} \right)^i \][/tex]
The geometric series sum formula for [tex]\( n \)[/tex] terms is:
[tex]\[ S_n = a \frac{1 - r^n}{1 - r} \][/tex]
where [tex]\( a = \frac{2}{3} \)[/tex], [tex]\( r = \frac{2}{3} \)[/tex], and [tex]\( n = 30 \)[/tex].
So, computing the series sum:
[tex]\[ S_{30} = \frac{2/3 \left(1 - \left(\frac{2}{3}\right)^{30}\right)}{1 - 2/3} \][/tex]
[tex]\[ S_{30} = 2 \left( \frac{1 - \left(\frac{2}{3}\right)^{30}}{1/3} \right) \][/tex]
[tex]\[ S_{30} = 6 \left( 1 - \left(\frac{2}{3}\right)^{30} \right) \][/tex]
The total distance for the bounces computation becomes:
[tex]\[ \text{Total Bounce Distance} = 2 \times 10 \times S_{30} = 20 \times 6 \left( 1 - \left(\frac{2}{3} \right)^{30} \right) \][/tex]
So, the total distance ball traveled:
[tex]\[ \approx 49.99979139619796 \text{ feet} \][/tex]
Thus, the total distance traveled by the ball is:
[tex]\[ 50 \, \text{feet} \][/tex] (rounded to the nearest integer).
So the total distance is approximately [tex]\( 5 \times 10^1 \)[/tex] feet in exponential notation.
