AW Score: $$ 71.67 \%, 14.33 $$ of 20 points A town currently has 4000 residents. The expected future population can be approximated by the function $$ \mathrm{P}(\mathrm{x})=4000(1.413)^{0.1 \mathrm{x}} $$ where x is the number of years in the future. Determine the expecte population of the town in 10 years and in 50 years. After 10 years, the population of the town will be (Round to the nearest whole number as needed.)

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Evaluate the expression by following steps:
- step0: Evaluate:
  $$4000\times 1.413^{0.1x}$$
- step1: Substitute:
  $$4000\times 1.413^{0.1\times 10}$$
- step2: Multiply the numbers:
  $$4000\times 1.413^{1}$$
- step3: Calculate:
  $$4000\times \frac{1413}{1000}$$
- step4: Reduce the numbers:
  $$4\times 1413$$
- step5: Multiply the numbers:
  $$5652$$
Substitute $$ x=50 $$ into the expression $$ 4000(1.413)^{0.1 x} $$.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$4000\times 1.413^{0.1x}$$
- step1: Substitute:
  $$4000\times 1.413^{0.1\times 50}$$
- step2: Multiply the numbers:
  $$4000\times 1.413^{5}$$
- step3: Convert the expressions:
  $$4000\left(\frac{1413}{1000}\right)^{5}$$
- step4: Simplify:
  $$4000\times \frac{1413^{5}}{1000^{5}}$$
- step5: Expand the expression:
  $$\frac{1413^{5}}{1024\times 125^{4}}$$
To determine the expected population of the town in 10 years and in 50 years, we will use the population function given by:

$$
P(x) = 4000(1.413)^{0.1x}
$$

### Step 1: Calculate the population in 10 years

Substituting $$ x = 10 $$:

$$
P(10) = 4000(1.413)^{0.1 \times 10} = 4000(1.413)^{1} = 4000 \times 1.413
$$

Calculating this gives:

$$
P(10) \approx 5652
$$

So, after 10 years, the population of the town will be approximately **5652** residents.

### Step 2: Calculate the population in 50 years

Substituting $$ x = 50 $$:

$$
P(50) = 4000(1.413)^{0.1 \times 50} = 4000(1.413)^{5}
$$

Calculating $$ (1.413)^{5} $$:

$$
(1.413)^{5} \approx 5.141
$$

Now substituting back:

$$
P(50) \approx 4000 \times 5.141 \approx 20564
$$

Thus, after 50 years, the expected population of the town will be approximately **20564** residents.

### Summary:
- Population in 10 years: **5652**
- Population in 50 years: **20564**
After 10 years, the town's population will be approximately 5,652 residents. After 50 years, it will be approximately 20,564 residents.

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