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)equalsminus4.182xsquaredplus72.85xplus96.42​, where xequals0 represents January​ 1, 2000, xequals1 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?

Answer 1

Answer:

This means that in the year 2008 the amount spent on clothing and footwear was a​ maximum.

Step-by-step explanation:

The quadratic function given is:

$f(x) = -4.182x^{2} + 72.85x + 96.42$

This is a quadratic function in the following format:

$f(x) = ax^{2} + bx + c$

When $a < 0$, as in this problem, the vortex is a maximum.

The vortex of the function is the point $(x_{v}, y_{v})$, in which

$x_{v} = -\frac{b}{2a}$

$y_{v} = f(x_{v})$

According to the​ model, in what year during this period was the amount spent on clothing and footwear a​ maximum?

The year is x in the function. The year in which the amount spent is a maximum is 2000(the initial year) added to $x_{v}$

In our secod order function, we have that $a = -4.182, b = 72.85$. So

$x_{v} = -\frac{72.85}{2*(-4.182)} = 8.71$

2000 + 8.71 = 2008.71.

This means that in the year 2008 the amount spent on clothing and footwear was a​ maximum.