Question

Solve the problem. In one U.S. city, the quadratic function f(x)= 0.039x²-0.48x+36.66 models the average age y, at which men were the 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 that men were first married that year? (Round to the nearest tenth.)

Answer 1

The average age at which men were first married was at a minimum in the year 1906, with an average age of 25.5 years.

Step-by-step explanation:

To find the year when the average age at which men were first married was at a minimum, we need to determine the x-value that corresponds to the vertex of the quadratic function. The vertex of a quadratic function in the form f(x) = ax² + bx + c can be found using the formula x = -b/2a.

In this case, the quadratic function is f(x) = 0.039x² - 0.48x + 36.66. So, a = 0.039 and b = -0.48. Plugging these values into the formula, we get x = -(-0.48)/(2*0.039) = 6.1538.

Since x represents the number of years after 1900, the year when the average age was at a minimum would be 1900 + 6.1538 = 1906.2. Rounded to the nearest year, the average age was at a minimum in the year 1906.

To find the average age that men were first married in that year, we substitute the x-value back into the quadratic function. f(6.1538) = 0.039*6.1538² - 0.48*6.1538 + 36.66 = 25.51. Rounded to the nearest tenth, the average age that men were first married in 1906 was 25.5 years.