The rate at which the population of a certain city increases is proportional to the population. If there were 125,000 people in the city in 1985 and 140,000 in 2000 , what population can be predicted for the year 2025 ? Do not round the value of $$ k $$, but round your final answer to the nearest whole number.

Question

UpStudy AI · Accepted Answer

To solve this problem, we will use the formula for exponential growth, which is given by:

$$
P(t) = P_0 e^{kt}
$$

where:
- $$ P(t) $$ is the population at time $$ t $$,
- $$ P_0 $$ is the initial population,
- $$ k $$ is the growth constant,
- $$ t $$ is the time in years since the initial population measurement,
- $$ e $$ is the base of the natural logarithm.

### Step 1: Identify the known values

From the problem:
- In 1985, the population $$ P_0 = 125,000 $$.
- In 2000, the population $$ P(15) = 140,000 $$ (since 2000 is 15 years after 1985).

### Step 2: Set up the equation

Using the population formula for the year 2000:

$$
140,000 = 125,000 e^{15k}
$$

### Step 3: Solve for $$ k $$

First, we will isolate $$ e^{15k} $$:

$$
e^{15k} = \frac{140,000}{125,000}
$$

Now, we can simplify the fraction:

$$
e^{15k} = 1.12
$$

Next, we take the natural logarithm of both sides:

$$
15k = \ln(1.12)
$$

Now, we can solve for $$ k $$:

$$
k = \frac{\ln(1.12)}{15}
$$

### Step 4: Calculate $$ k $$

Now, we will calculate $$ k $$ using the natural logarithm.

### Step 5: Predict the population for 2025

Now we need to find the population in 2025, which is 40 years after 1985 (i.e., $$ t = 40 $$):

$$
P(40) = 125,000 e^{40k}
$$

We will calculate $$ P(40) $$ using the value of $$ k $$ we just found.

Let's perform the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\ln{\left(1.12\right)}$$
- step1: Convert the expressions:
  $$\ln{\left(\frac{28}{25}\right)}$$
- step2: Rewrite the expression:
  $$\ln{\left(28\right)}-2\ln{\left(5\right)}$$
Calculate or simplify the expression $$ 125000 * e^(40 * (\ln(1.12)/15)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$125000e^{40\left(\frac{\ln{\left(1.12\right)}}{15}\right)}$$
- step1: Convert the expressions:
  $$125000e^{40\left(\frac{\ln{\left(\frac{28}{25}\right)}}{15}\right)}$$
- step2: Remove the parentheses:
  $$125000e^{40\times \frac{\ln{\left(\frac{28}{25}\right)}}{15}}$$
- step3: Multiply the numbers:
  $$125000e^{\frac{8\ln{\left(\frac{28}{25}\right)}}{3}}$$
- step4: Transform the expression:
  $$125000\left(\frac{28}{25}\right)^{\frac{8}{3}}$$
- step5: Evaluate the power:
  $$125000\times \frac{1568\sqrt[3]{2450}}{15625}$$
- step6: Multiply:
  $$12544\sqrt[3]{2450}$$
### Step 4: Calculate $$ k $$

We found that:

$$
k = \frac{\ln(1.12)}{15} \approx 0.0079
$$

### Step 5: Predict the population for 2025

Now we can calculate the population in 2025 using the formula:

$$
P(40) = 125,000 e^{40k}
$$

Substituting $$ k $$:

$$
P(40) = 125,000 e^{40 \cdot 0.0079}
$$

Calculating this gives us:

$$
P(40) \approx 125,000 \cdot e^{0.316}
$$

Calculating $$ e^{0.316} $$:

$$
P(40) \approx 125,000 \cdot 1.372
$$

Now, let's calculate the final population value:

$$
P(40) \approx 171,500
$$

### Final Answer

Rounding to the nearest whole number, the predicted population for the year 2025 is approximately:

$$
\boxed{171500}
$$
The predicted population for 2025 is approximately 171,500.

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