Choose the one alternative that best completes the statement or answers the question.
-The quadratic function $f ( x ) = 0.0040 x ^ { 2 } - 0.43 x + 36.98$ models the median, or average, age, $y$, at which U.S. men were first married x years after 1900 . In which year was this average age at a minimum? (Round to the nearest year.) What was the average age at first marriage for that year? (Round to the nearest tenth.)
A) 1953,36 years old
B) $1936,48.5$ years old
C) $1954,25.4$ years old
D) $1954,48.5$ years old

Question

Quizplus · Accepted Answer

To find the year when the average age at first marriage was at a minimum, we need to find the vertex of the given quadratic function. The x-coordinate of the vertex can be found using the formula $-\frac{b}{2a}$, where $a = 0.0040$ and $b = -0.43$. Plugging these values in gives $-\frac{-0.43}{2(0.0040)} = 53.75$. Since $x$ represents years after 1900, we add this to 1900 to get 1953.75, which rounds to 1954. To find the minimum average age, we substitute $x = 53.75$ into the original equation to get $f(53.75) = 0.0040(53.75)^2 - 0.43(53.75) + 36.98 \approx 25.4$. Therefore, the correct answer is 1954, with an average age of 25.4 years.

Choose the One Alternative That Best Completes the Statement or Answers