Question

The total amount spent by some number of people on clothing and footwear in the years 2000-2009 can be modeled by the quadratic function f(x)=-4.353x² + 71.52x + 96.26, where x=0 represents January 1, 2000, x=1 represents January 1, 2001, and so on, and f(x) is in billions of dollars. According to the model, in what year during this period was the amount spent on clothing and footwear a maximum? In the year , $ billion was spent on clothing and footwear.

a) 2002, $125.89 billion
b) 2004, $112.43 billion
c) 2006, $99.54 billion
d) 2008, $86.98 billion

Answer 1

To find the year when the amount spent on clothing and footwear was a maximum, substitute the x-coordinate of the vertex into the equation. The maximum amount spent was $86.98 billion in the year 2008.

Step-by-step explanation:

To find the year when the amount spent on clothing and footwear was a maximum, 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 is given by the coordinates (-b/2a, f(-b/2a)).

In this case, the quadratic function is f(x) = -4.353x^2 + 71.52x + 96.26. By substituting the values into the formula, we obtain the vertex coordinates (-71.52/(2*-4.353), f(-71.52/(2*-4.353))). Simplifying the expression gives (8.218, f(8.218)).

Therefore, the year when the amount spent on clothing and footwear was a maximum is 2008. We can find the amount spent by substituting x = 8 into the equation f(x) = -4.353x^2 + 71.52x + 96.26. The result is $86.98 billion.