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Answer :
To solve the problem, let's think about each bounce of the bouncy ball.
1. Initial Drop: The ball is first dropped from a height of 193.00 cm.
2. Understanding the Bounce Reduction: After each bounce, the ball reaches a height that is 17% less than the previous height. This means it retains 83% (since 100% - 17% = 83%) of the height from the previous bounce. We can express 83% as the decimal 0.83.
3. Finding the Height After Each Bounce: We want to calculate the height reached after the 10th bounce. To find this, we multiply the initial height by 0.83 for each bounce. This can be described by the formula for exponential decay:
[tex]\[
\text{Height after } n \text{ bounces} = 193 \times (0.83)^n
\][/tex]
We are interested in the height after the 10th bounce, so substitute [tex]\( n = 10 \)[/tex]:
[tex]\[
\text{Height after 10 bounces} = 193 \times (0.83)^{10}
\][/tex]
4. Calculating the Height: Now, using the provided computations, the height after the 10th bounce is approximately:
[tex]\[
29.95 \text{ cm}
\][/tex]
Therefore, the maximum height reached after the 10th bounce is approximately 29.95 cm. Thus, the correct choice from the options given is:
D) [tex]\(193 \cdot 0.83^{10} \approx 29.95 \text{ cm}\)[/tex]
1. Initial Drop: The ball is first dropped from a height of 193.00 cm.
2. Understanding the Bounce Reduction: After each bounce, the ball reaches a height that is 17% less than the previous height. This means it retains 83% (since 100% - 17% = 83%) of the height from the previous bounce. We can express 83% as the decimal 0.83.
3. Finding the Height After Each Bounce: We want to calculate the height reached after the 10th bounce. To find this, we multiply the initial height by 0.83 for each bounce. This can be described by the formula for exponential decay:
[tex]\[
\text{Height after } n \text{ bounces} = 193 \times (0.83)^n
\][/tex]
We are interested in the height after the 10th bounce, so substitute [tex]\( n = 10 \)[/tex]:
[tex]\[
\text{Height after 10 bounces} = 193 \times (0.83)^{10}
\][/tex]
4. Calculating the Height: Now, using the provided computations, the height after the 10th bounce is approximately:
[tex]\[
29.95 \text{ cm}
\][/tex]
Therefore, the maximum height reached after the 10th bounce is approximately 29.95 cm. Thus, the correct choice from the options given is:
D) [tex]\(193 \cdot 0.83^{10} \approx 29.95 \text{ cm}\)[/tex]
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