Question
A boat washes up onto an island, introducing 25 rabbits into the environment. The rabbits multiply at a rate of 50 percent every month. Does this graph accurately model the situation? Enter 1 for yes or 2 for no.
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Answers
Answered by
GPT-4o mini
To determine if the graph accurately models the situation, we need to consider the growth rate of the rabbit population. The initial population is 25 rabbits, and they multiply at a rate of 50 percent every month.
The population at the end of each month can be modeled with the formula:
\[ P(t) = P_0 \times (1 + r)^t \]
Where:
- \( P_0 \) is the initial population (25 rabbits),
- \( r \) is the growth rate (0.50 for 50%),
- \( t \) is the time in months.
After 1 month, the population would be:
\[ P(1) = 25 \times (1 + 0.5)^1 = 25 \times 1.5 = 37.5 \text{ rabbits} \] (we usually round to nearest whole numbers)
After 2 months:
\[ P(2) = 25 \times (1 + 0.5)^2 = 25 \times 1.5^2 = 25 \times 2.25 = 56.25 \text{ rabbits} \]
And so on.
The correct mathematical modeling would show an exponential growth, and the shape of the graph should be an upward curve that becomes steeper over time.
If the graph you are responding to reflects this growth pattern (i.e., exponential growth with a continuously increasing curve), then you would enter 1 for yes. If it shows linear growth or any other incorrect pattern, you would enter 2 for no.
Since I cannot see the graph, you will have to determine if it matches the description above. Based on the information given, I would suggest:
- If the graph is exponential growth: Enter **1**.
- If the graph does not represent exponential growth: Enter **2**.
The population at the end of each month can be modeled with the formula:
\[ P(t) = P_0 \times (1 + r)^t \]
Where:
- \( P_0 \) is the initial population (25 rabbits),
- \( r \) is the growth rate (0.50 for 50%),
- \( t \) is the time in months.
After 1 month, the population would be:
\[ P(1) = 25 \times (1 + 0.5)^1 = 25 \times 1.5 = 37.5 \text{ rabbits} \] (we usually round to nearest whole numbers)
After 2 months:
\[ P(2) = 25 \times (1 + 0.5)^2 = 25 \times 1.5^2 = 25 \times 2.25 = 56.25 \text{ rabbits} \]
And so on.
The correct mathematical modeling would show an exponential growth, and the shape of the graph should be an upward curve that becomes steeper over time.
If the graph you are responding to reflects this growth pattern (i.e., exponential growth with a continuously increasing curve), then you would enter 1 for yes. If it shows linear growth or any other incorrect pattern, you would enter 2 for no.
Since I cannot see the graph, you will have to determine if it matches the description above. Based on the information given, I would suggest:
- If the graph is exponential growth: Enter **1**.
- If the graph does not represent exponential growth: Enter **2**.
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