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Suppose that the population of a certain city during a certain time period can be approximated by \( P(x)=-0.1 x^{5}+3.7 x^{4}+6,000 \), where \( x \) is time in years since 1960. Find \( P(5) \). A. 7,100 B. 10,800 C. 8,000 D. 12,500

Ask by Cross Gardner. in the United States
Mar 20,2025

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\( P(5) = 8,000 \), so the answer is C. 8,000.

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To find \( P(5) \), we substitute \( x = 5 \) into the population function: \[ P(5) = -0.1(5)^{5} + 3.7(5)^{4} + 6,000 \] Calculating each term: - \( (5)^{5} = 3125 \), so \( -0.1(3125) = -312.5 \) - \( (5)^{4} = 625 \), so \( 3.7(625) = 2312.5 \) Now, substitute these values into the equation: \[ P(5) = -312.5 + 2312.5 + 6000 \] \[ P(5) = 6000 + 2000 = 8000 \] Thus, \( P(5) = 8000 \), which corresponds to option C.

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