Questions: The exponential model A = 634.0(1.16)t describes the population, A, of a county in millions, t years after 2003. Use this model to determine the population of the county in 2003.

The exponential model A = 634.0(1.16)t describes the population, A, of a county in millions, t years after 2003. Use this model to determine the population of the county in 2003.
Transcript text: The exponential model A = 634.0(1.16)t describes the population, A, of a county in millions, t years after 2003. Use this model to determine the population of the county in 2003.
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Solution

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To determine the population of the county in 2003 using the given exponential model \( A = 634.0(1.16)^t \), we need to evaluate the model at \( t = 0 \) because 2003 is the starting year. This will give us the initial population of the county in 2003.

Step 1: Identify the Model

The exponential model for the population \( A \) of the county is given by the equation: \[ A = 634.0(1.16)^t \] where \( t \) represents the number of years after 2003.

Step 2: Evaluate the Model for 2003

To find the population in the year 2003, we set \( t = 0 \): \[ A = 634.0(1.16)^0 \] Since any number raised to the power of 0 is 1, we simplify this to: \[ A = 634.0 \times 1 = 634.0 \]

Step 3: State the Population

Thus, the population of the county in 2003 was: \[ \boxed{634.0} \]

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