To find the year when the average age at first marriage is at a minimum, and to determine what that minimum age was, we need to analyze the given quadratic function:
[tex]\[ f(x) = 0.0037x^2 - 0.45x + 36.29 \][/tex]
This function models the median age, [tex]\( y \)[/tex], at which men were first married, [tex]\( x \)[/tex] years after 1900.
### Step-by-Step Solution:
1. Identify the form of the quadratic function:
The function [tex]\( f(x) = 0.0037x^2 - 0.45x + 36.29 \)[/tex] is in the standard form [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\( a = 0.0037 \)[/tex]
- [tex]\( b = -0.45 \)[/tex]
- [tex]\( c = 36.29 \)[/tex]
2. Determine the x-coordinate of the vertex:
The minimum or maximum value of a quadratic function occurs at its vertex. The x-coordinate of the vertex for a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
3. Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ x = -\frac{-0.45}{2 \times 0.0037} \][/tex]
[tex]\[ x = \frac{0.45}{0.0074} \][/tex]
[tex]\[ x \approx 60.81081 \][/tex]
4. Convert x years after 1900 to an actual year:
Since [tex]\( x \)[/tex] is the number of years after 1900, we add 1900 to the x-coordinate of the vertex:
[tex]\[ \text{Year} = 1900 + 60.81081 \][/tex]
[tex]\[ \text{Year} \approx 1961 \][/tex]
(when rounded to the nearest year).
5. Calculate the average age at the vertex:
To find the minimum average age, substitute [tex]\( x = 60.81081 \)[/tex] back into the quadratic function:
[tex]\[ f(60.81081) = 0.0037 \times (60.81081)^2 - 0.45 \times 60.81081 + 36.29 \][/tex]
[tex]\[ f(60.81081) \approx 22.6 \][/tex]
(when rounded to the nearest tenth).
### Conclusion:
The year when the average age at first marriage is at a minimum is approximately 1961, and the average age at first marriage for that year is approximately 22.6 years.
Thus, the correct answer is:
Option A. 1960, 22.6 years old
