Thank you for visiting A video game tournament begins with 625 competitors After each round one fifth of the competitors remain Write the function that models the change between. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
Sure! Let's tackle this problem step-by-step.
We have a video game tournament with 625 competitors at the beginning. Each round reduces the number of competitors to one-fifth of the previous number. We need to express the number of competitors remaining after each round, using a function of the number of rounds, [tex]\( x \)[/tex].
### Step-by-Step Solution:
1. Initial Situation:
- At round 0 (before any elimination), there are 625 competitors. Thus, initially, we have [tex]\( r(0) = 625 \)[/tex].
2. Competitors Remaining After Each Round:
- After the first round ([tex]\( x = 1 \)[/tex]), only one-fifth of the competitors remain. Thus [tex]\( r(1) = \frac{1}{5} \times 625 \)[/tex].
- After the second round ([tex]\( x = 2 \)[/tex]), one-fifth of the competitors from the first round remains. Thus [tex]\( r(2) = \left(\frac{1}{5}\right)^2 \times 625 \)[/tex].
- This pattern continues. After [tex]\( x \)[/tex] rounds, the remaining competitors are [tex]\( r(x) = 625 \times \left(\frac{1}{5}\right)^x \)[/tex].
3. Exponential Function:
- The expression [tex]\( r(x) = 625 \times \left(\frac{1}{5}\right)^x \)[/tex] is an exponential function. Here, [tex]\( 625 \)[/tex] is the initial number of competitors, and [tex]\( \left(\frac{1}{5}\right)^x \)[/tex] represents the fraction remaining after [tex]\( x \)[/tex] rounds.
4. Match With Provided Options:
- We need to choose the correct function from the given options:
- [tex]\( r(x) = 625 \times 5^x \)[/tex] is incorrect as it increases the number of competitors.
- [tex]\( r(x) = 5 \times 625^x \)[/tex] doesn't fit the situation.
- [tex]\( n(x) = 5 \times\left(\frac{1}{625}\right)^x \)[/tex] doesn't make sense contextually.
- [tex]\( r(x) = 625 \times\left(\frac{1}{5}\right)^x \)[/tex] matches our derived function.
### Conclusion:
The correct function that models the number of competitors as a function of the number of rounds is:
[tex]\[ r(x) = 625 \times \left(\frac{1}{5}\right)^x \][/tex]
This represents how the number of competitors decreases by a factor of [tex]\( \frac{1}{5} \)[/tex] each round, starting with 625 competitors.
We have a video game tournament with 625 competitors at the beginning. Each round reduces the number of competitors to one-fifth of the previous number. We need to express the number of competitors remaining after each round, using a function of the number of rounds, [tex]\( x \)[/tex].
### Step-by-Step Solution:
1. Initial Situation:
- At round 0 (before any elimination), there are 625 competitors. Thus, initially, we have [tex]\( r(0) = 625 \)[/tex].
2. Competitors Remaining After Each Round:
- After the first round ([tex]\( x = 1 \)[/tex]), only one-fifth of the competitors remain. Thus [tex]\( r(1) = \frac{1}{5} \times 625 \)[/tex].
- After the second round ([tex]\( x = 2 \)[/tex]), one-fifth of the competitors from the first round remains. Thus [tex]\( r(2) = \left(\frac{1}{5}\right)^2 \times 625 \)[/tex].
- This pattern continues. After [tex]\( x \)[/tex] rounds, the remaining competitors are [tex]\( r(x) = 625 \times \left(\frac{1}{5}\right)^x \)[/tex].
3. Exponential Function:
- The expression [tex]\( r(x) = 625 \times \left(\frac{1}{5}\right)^x \)[/tex] is an exponential function. Here, [tex]\( 625 \)[/tex] is the initial number of competitors, and [tex]\( \left(\frac{1}{5}\right)^x \)[/tex] represents the fraction remaining after [tex]\( x \)[/tex] rounds.
4. Match With Provided Options:
- We need to choose the correct function from the given options:
- [tex]\( r(x) = 625 \times 5^x \)[/tex] is incorrect as it increases the number of competitors.
- [tex]\( r(x) = 5 \times 625^x \)[/tex] doesn't fit the situation.
- [tex]\( n(x) = 5 \times\left(\frac{1}{625}\right)^x \)[/tex] doesn't make sense contextually.
- [tex]\( r(x) = 625 \times\left(\frac{1}{5}\right)^x \)[/tex] matches our derived function.
### Conclusion:
The correct function that models the number of competitors as a function of the number of rounds is:
[tex]\[ r(x) = 625 \times \left(\frac{1}{5}\right)^x \][/tex]
This represents how the number of competitors decreases by a factor of [tex]\( \frac{1}{5} \)[/tex] each round, starting with 625 competitors.
Thank you for reading the article A video game tournament begins with 625 competitors After each round one fifth of the competitors remain Write the function that models the change between. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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