The current population of a town is 10,000. If the population, P, increase by 3.5% every six months, which equation could be used to find the population after t years?

Options

A. `P=10,000(1.035)^((t)/(2))`
B. `P=10,000(0.965)^(2t)`
C. `P=10,000(1.035)^(2t)`
D. `P=10,000(0.965)^((t)/(2))`

Answer

To find the population of the town after \( t \) years, given that the current population is 10,000 and it increases by 3.5% every six months, we can follow these steps:

### Step 1: Understand the Initial Population
The current population of the town is given as:
\[ P_0 = 10,000 \]

### Step 2: Determine the Growth Rate
The population increases by 3.5% every six months. This means that after each six-month period, the population is multiplied by:
\[ 1 + \frac{3.5}{100} = 1.035 \]

### Step 3: Calculate the Population After One Year
Since there are two six-month periods in one year, the population after one year can be calculated as:
\[ P(1) = P_0 \times (1.035)^2 \]
Substituting the initial population:
\[ P(1) = 10,000 \times (1.035)^2 \]

### Step 4: Generalize for \( t \) Years
For \( t \) years, there will be \( 2t \) six-month periods. Therefore, the population after \( t \) years can be expressed as:
\[ P(t) = P_0 \times (1.035)^{2t} \]
Substituting the initial population:
\[ P(t) = 10,000 \times (1.035)^{2t} \]

### Final Equation
Thus, the equation that can be used to find the population after \( t \) years is:
\[ P(t) = 10,000 \times (1.035)^{2t} \]

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