In one U.S. city, the quadratic function models the median, or average, age, y, at which 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.)
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What quadratic function?
To find the year in which the average age at first marriage was at a minimum, we need to determine the vertex of the quadratic function. The vertex of a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b / (2a).
In this case, the quadratic function is given as y = 0.003x^2 - 0.6x + 56, where x represents the number of years after 1900 and y represents the average age at first marriage.
Comparing this function to the general form ax^2 + bx + c, we can see that a = 0.003, b = -0.6, and c = 56.
Using the formula x = -b / (2a), we can calculate the x-coordinate of the vertex.
x = -( -0.6 ) / ( 2 * 0.003 )
x = 0.6 / 0.006
x = 100
Therefore, the x-coordinate of the vertex is 100 years after 1900, which corresponds to the year 2000.
To find the average age at first marriage for that year, we substitute the value of x = 100 into the function y = 0.003x^2 - 0.6x + 56.
y = 0.003(100)^2 - 0.6(100) + 56
y = 0.003(10000) - 60 + 56
y = 30 - 60 + 56
y = 26
Therefore, the average age at first marriage in the year 2000 is approximately 26 years.
To determine the year at which the average age at first marriage was at a minimum, we need to analyze the quadratic function and find the x-coordinate of the vertex. The vertex form of a quadratic function is given by:
f(x) = a(x - h)^2 + k
In this case, the function is given by y = ax^2 + bx + c, where y represents the average age at first marriage and x represents the number of years after 1900. To find the year at which the average age is at a minimum, we need to find the value of x when the quadratic function is at its minimum.
The x-coordinate of the vertex of a quadratic function in the form y = ax^2 + bx + c is given by:
x = -b / (2a)
Let's find the x-coordinate of the vertex for the given quadratic function, y = ax^2 + bx + c.
Since the function is not provided, we need the values of a, b, and c in order to proceed with the calculations. Please provide the values of a, b, and c.