In one U.S. city, the quadratic function [tex]f(x) = 0.0037x^2 - 0.45x + 36.29[/tex] models the median, or average, age, [tex]y[/tex], at which men were first married [tex]x[/tex] years after 1900.

In which year was the 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. 1960, 22.6 years old
B. 1936, 50.0 years old
C. 1952, 36.0 years old
D. 1960, 50.0 years old



Answer :

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

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