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
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
Solution
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.
Let the population 5 years ago be P0.
Given that the current population P is 807, and it is 3 times what it was 5 years ago, we can write 807=3×P0.
Solve for P0.
Use the exponential growth formula P(t)=P0×ekt to find the growth rate k.
Use the growth rate k to predict the population 7 years from now.
Step 1: Calculate Initial Population
Given that the current population P=807 is 3 times the population 5 years ago, we can express this as:
P0=3P=3807=269
Step 2: Determine Growth Rate
Using the exponential growth formula P=P0ekt, we can find the growth rate k. From the equation:
807=269e5k
we can rearrange it to:
e5k=269807=3
Taking the natural logarithm of both sides gives:
5k=ln(3)⟹k=5ln(3)≈0.2197
Step 3: Predict Future Population
To find the population 7 years from now, we use the formula again:
P(t)=P0ekt
where t=5+7=12:
P(12)=269e12k=269e12⋅0.2197≈3757.0181
Final Answer
The population 7 years from now will be approximately 3757.