The population, \( p \), of the United States in 2004 was approximately 295 million. If \( p \) grows at a rate proportional to itself, with \( k=0.017 \) (or 1.7 percent per year), calculate the value of \( p \) in 2007 . Hint: \( \frac{d p}{d t}=k p \) Do not round the value of \( k \), but round your final answer to the nearest million people. \[ p=[?] \text { million people } \]

To solve this problem, we can use the formula for exponential growth given by the differential equation \( \frac{d p}{d t} = k p \). The solution to this equation can be expressed as: \[ p(t) = p_0 e^{kt} \] where: - \( p_0 \) is the initial population (in this case, 295 million in 2004), - \( k \) is the growth rate (0.017), - \( t \) is the number of years after 2004. Calculating \( t \) for the year 2007, we have: \[ t = 2007 - 2004 = 3 \text{ years} \] Now substituting the values into the formula gives: \[ p(3) = 295 e^{0.017 \cdot 3} \] Calculating \( 0.017 \cdot 3 = 0.051 \), we have: \[ p(3) = 295 e^{0.051} \] Using a calculator or mathematical software, we find \( e^{0.051} \approx 1.0522 \). Now substituting this value back in gives: \[ p(3) \approx 295 \cdot 1.0522 \approx 310.63 \] Rounding to the nearest million, the value of \( p \) in 2007 is: \[ p \approx 311 \text{ million people.} \]