Questions: Question 8 (1 point) The population of a small village is now 807, which is 3 times what it was 5 years ago. If the population continues to increase exponentially at this rate, then the population 7 years from now will be 3757 1560 2905 4842

Question 8 (1 point)
The population of a small village is now 807, which is 3 times what it was 5 years ago. If the population continues to increase exponentially at this rate, then the population 7 years from now will be
3757
1560
2905
4842
Transcript text: Question 8 (1 point) The population of a small village is now 807, which is 3 times what it was 5 years ago. If the population continues to increase exponentially at this rate, then the population 7 years from now will be 3757 1560 2905 4842
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Solution

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Solution Steps

To solve this problem, we need to determine the exponential growth rate of the population and then use it to predict the population 7 years from now.

  1. Let the population 5 years ago be P0 P_0 .
  2. Given that the current population P P is 807, and it is 3 times what it was 5 years ago, we can write 807=3×P0 807 = 3 \times P_0 .
  3. Solve for P0 P_0 .
  4. Use the exponential growth formula P(t)=P0×ekt P(t) = P_0 \times e^{kt} to find the growth rate k k .
  5. Use the growth rate k k to predict the population 7 years from now.
Step 1: Calculate Initial Population

Given that the current population P=807 P = 807 is 3 times the population 5 years ago, we can express this as: P0=P3=8073=269 P_0 = \frac{P}{3} = \frac{807}{3} = 269

Step 2: Determine Growth Rate

Using the exponential growth formula P=P0ekt P = P_0 e^{kt} , we can find the growth rate k k . From the equation: 807=269e5k 807 = 269 e^{5k} we can rearrange it to: e5k=807269=3 e^{5k} = \frac{807}{269} = 3 Taking the natural logarithm of both sides gives: 5k=ln(3)    k=ln(3)50.2197 5k = \ln(3) \implies k = \frac{\ln(3)}{5} \approx 0.2197

Step 3: Predict Future Population

To find the population 7 years from now, we use the formula again: P(t)=P0ekt P(t) = P_0 e^{kt} where t=5+7=12 t = 5 + 7 = 12 : P(12)=269e12k=269e120.21973757.0181 P(12) = 269 e^{12k} = 269 e^{12 \cdot 0.2197} \approx 3757.0181

Final Answer

The population 7 years from now will be approximately 3757 \boxed{3757} .

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