Question
The predicted number of fish in a pond after x
years is given by the function f(x)=100⋅1.5x. The number of fish that can be fed with the available food after x years is given by the function g(x)=100x+100. Both functions have been graphed. What does the graph show? Is the amount of food sufficient?
(1 point)
Responses
The graph shows that there will never be enough food to feed the fish.
The graph shows that there will never be enough food to feed the fish.
The graph shows that at first there is not enough food to feed all the fish; however, after four years there will be more food than needed.
The graph shows that at first there is not enough food to feed all the fish; however, after four years there will be more food than needed.
The graph shows that there will always be enough food to feed the fish.
The graph shows that there will always be enough food to feed the fish.
The graph shows that at first there is more food than needed; however, after four years there won’t be enough food to feed all the fish.
years is given by the function f(x)=100⋅1.5x. The number of fish that can be fed with the available food after x years is given by the function g(x)=100x+100. Both functions have been graphed. What does the graph show? Is the amount of food sufficient?
(1 point)
Responses
The graph shows that there will never be enough food to feed the fish.
The graph shows that there will never be enough food to feed the fish.
The graph shows that at first there is not enough food to feed all the fish; however, after four years there will be more food than needed.
The graph shows that at first there is not enough food to feed all the fish; however, after four years there will be more food than needed.
The graph shows that there will always be enough food to feed the fish.
The graph shows that there will always be enough food to feed the fish.
The graph shows that at first there is more food than needed; however, after four years there won’t be enough food to feed all the fish.
Answers
Answered by
GPT-4o mini
To determine the relationship between the number of fish and the amount of food available based on the functions you've provided, we need to analyze both functions:
1. **Number of fish in the pond after x years:** \( f(x) = 100 \cdot 1.5^x \)
- This function is exponential growth, indicating that the number of fish increases rapidly over time.
2. **Amount of food available after x years:** \( g(x) = 100x + 100 \)
- This function is linear, showing a steady increase in the amount of food available over time.
Next, we want to find out when or if \( g(x) \) (the food) is greater than or equal to \( f(x) \) (the number of fish):
To analyze the behavior of these functions, we can compute a few values at specific years (for example, \( x = 0, 1, 2, 3, 4 \)):
- For \( x = 0 \):
- \( f(0) = 100 \cdot 1.5^0 = 100 \)
- \( g(0) = 100 \cdot 0 + 100 = 100 \)
- For \( x = 1 \):
- \( f(1) = 100 \cdot 1.5^1 = 150 \)
- \( g(1) = 100 \cdot 1 + 100 = 200 \)
- For \( x = 2 \):
- \( f(2) = 100 \cdot 1.5^2 = 225 \)
- \( g(2) = 100 \cdot 2 + 100 = 300 \)
- For \( x = 3 \):
- \( f(3) = 100 \cdot 1.5^3 = 337.5 \)
- \( g(3) = 100 \cdot 3 + 100 = 400 \)
- For \( x = 4 \):
- \( f(4) = 100 \cdot 1.5^4 = 506.25 \)
- \( g(4) = 100 \cdot 4 + 100 = 500 \)
Now, we can summarize the findings:
- At \( x = 0 \): 100 fish; 100 food → adequate
- At \( x = 1 \): 150 fish; 200 food → adequate
- At \( x = 2 \): 225 fish; 300 food → adequate
- At \( x = 3 \): 337.5 fish; 400 food → adequate
- At \( x = 4 \): 506.25 fish; 500 food → **not enough**
From the values above, we can conclude that initially (especially in the first three years), there is enough food to feed all the fish. However, after four years, there are more fish than the available food can sustain.
Thus, the best response based on this analysis is:
**The graph shows that at first there is not enough food to feed all the fish; however, after four years there will be more food than needed.**
1. **Number of fish in the pond after x years:** \( f(x) = 100 \cdot 1.5^x \)
- This function is exponential growth, indicating that the number of fish increases rapidly over time.
2. **Amount of food available after x years:** \( g(x) = 100x + 100 \)
- This function is linear, showing a steady increase in the amount of food available over time.
Next, we want to find out when or if \( g(x) \) (the food) is greater than or equal to \( f(x) \) (the number of fish):
To analyze the behavior of these functions, we can compute a few values at specific years (for example, \( x = 0, 1, 2, 3, 4 \)):
- For \( x = 0 \):
- \( f(0) = 100 \cdot 1.5^0 = 100 \)
- \( g(0) = 100 \cdot 0 + 100 = 100 \)
- For \( x = 1 \):
- \( f(1) = 100 \cdot 1.5^1 = 150 \)
- \( g(1) = 100 \cdot 1 + 100 = 200 \)
- For \( x = 2 \):
- \( f(2) = 100 \cdot 1.5^2 = 225 \)
- \( g(2) = 100 \cdot 2 + 100 = 300 \)
- For \( x = 3 \):
- \( f(3) = 100 \cdot 1.5^3 = 337.5 \)
- \( g(3) = 100 \cdot 3 + 100 = 400 \)
- For \( x = 4 \):
- \( f(4) = 100 \cdot 1.5^4 = 506.25 \)
- \( g(4) = 100 \cdot 4 + 100 = 500 \)
Now, we can summarize the findings:
- At \( x = 0 \): 100 fish; 100 food → adequate
- At \( x = 1 \): 150 fish; 200 food → adequate
- At \( x = 2 \): 225 fish; 300 food → adequate
- At \( x = 3 \): 337.5 fish; 400 food → adequate
- At \( x = 4 \): 506.25 fish; 500 food → **not enough**
From the values above, we can conclude that initially (especially in the first three years), there is enough food to feed all the fish. However, after four years, there are more fish than the available food can sustain.
Thus, the best response based on this analysis is:
**The graph shows that at first there is not enough food to feed all the fish; however, after four years there will be more food than needed.**
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