16 Assume that the current population of a large region is 340 million. It was reported that the population grew since it was last measured 10 years previously, Use the model . Complete
parts a through c below.
a. What was the population when it was measured 10 years ago?
The population was million the first time it was measured.
(Round to one decimal place as needed.)

Ask by Fitzgerald Rowe. in the United States

Nov 05,2024

Question

16 Assume that the current population of a large region is 340 million. It was reported that the population grew since it was last measured 10 years previously, Use the model . Complete
parts a through c below.
a. What was the population when it was measured 10 years ago?
The population was million the first time it was measured.
(Round to one decimal place as needed.)

Ask by Fitzgerald Rowe. in the United States

Nov 05,2024

UpStudy AI · Accepted Answer

Given:
- Current population of the region: 340 million
- Population growth rate: 10.4%
- Time since last measurement: 10 years
- Model for population growth: $$ A = A_0 e^{kt} $$

We need to find the population when it was measured 10 years ago.

Let's denote:
- $$ A_0 $$ as the population 10 years ago
- $$ A $$ as the current population (340 million)
- $$ k $$ as the growth rate constant
- $$ t $$ as the time since last measurement (10 years)

From the model $$ A = A_0 e^{kt} $$, we can rearrange it to solve for $$ A_0 $$:
$$ A_0 = \frac{A}{e^{kt}} $$

Given that the population grew by 10.4% in 10 years, we can express this as:
$$ A = A_0 + 0.104 \times A_0 = 1.104 \times A_0 $$

Substitute the values of $$ A $$ and $$ t $$ into the equation:
$$ 340 = 1.104 \times A_0 $$

Now, we can solve for $$ A_0 $$ by dividing both sides by 1.104:
$$ A_0 = \frac{340}{1.104} $$

Let's calculate the value of $$ A_0 $$ to find the population 10 years ago.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{340}{1.104}$$
- step1: Convert the expressions:
  $$\frac{340}{\frac{138}{125}}$$
- step2: Multiply by the reciprocal:
  $$340\times \frac{125}{138}$$
- step3: Reduce the numbers:
  $$170\times \frac{125}{69}$$
- step4: Multiply:
  $$\frac{170\times 125}{69}$$
- step5: Multiply:
  $$\frac{21250}{69}$$
The population 10 years ago was approximately 307.97 million.

Therefore, the population was 307.97 million the first time it was measured.
The population 10 years ago was approximately 308 million.

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