Questions: The per capita (per person) income from 1980 to 2010 can be modeled by f(x)=1000(x-1980)+15,000 where x is the year. Determine the year when the per capita income was 24,000. The per capita income was 24,000 in the year

The per capita (per person) income from 1980 to 2010 can be modeled by f(x)=1000(x-1980)+15,000 where x is the year. Determine the year when the per capita income was 24,000. The per capita income was 24,000 in the year
Transcript text: The per capita (per person) income from 1980 to 2010 can be modeled by \[ f(x)=1000(x-1980)+15,000 \] where $x$ is the year. Determine the year when the per capita income was $\$ 24,000$. The per capita income was $\$ 24,000$ in the year $\square$
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Solution

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Solution Steps

To determine the year when the per capita income was $24,000, we need to solve the equation \( f(x) = 24,000 \) for \( x \). Given the function \( f(x) = 1000(x - 1980) + 15,000 \), we can set it equal to $24,000 and solve for \( x \).

Step 1: Set up the equation

We are given the function for per capita income: \[ f(x) = 1000(x - 1980) + 15,000 \] We need to determine the year \( x \) when the per capita income was \$24,000. Therefore, we set up the equation: \[ 1000(x - 1980) + 15,000 = 24,000 \]

Step 2: Solve for \( x \)

First, subtract 15,000 from both sides of the equation: \[ 1000(x - 1980) = 24,000 - 15,000 \] \[ 1000(x - 1980) = 9,000 \]

Next, divide both sides by 1000: \[ x - 1980 = \frac{9,000}{1000} \] \[ x - 1980 = 9 \]

Finally, add 1980 to both sides to solve for \( x \): \[ x = 9 + 1980 \] \[ x = 1989 \]

Final Answer

\(\boxed{x = 1989}\)

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