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. 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 of clothing and footwear a maximum In the year (blank) , (blank) billion was spent on clothing and footwear.

Answer 1

For this problem, we are given a quadratic equation that models the total amount spent on clothing and footwear in the years 2000-2009. We need to use the model to determine the maximum amount spent during the period.

The equation is shown below:

$f(x)=-4.462x^2+71.54x+97.44$

Since the leading term is negative, the vertex of this function will represent an absolute maximum value. Therefore we can determine the vertex to answer the problem, the vertex's coordinates are given below:

$\begin{gathered} x_{max}=\frac{-b}{2a}\\ \\ y_{max}=f(x_{max}) \end{gathered}$

Then we have:

$\begin{gathered} x_{max}=\frac{-71.54}{2\cdot(-4.462)}=8.017\\ \\ y_{max}=-4.462\cdot(8.017)^2+71.54\cdot(8.017)+97.44=384.2 \end{gathered}$

In the year 2008, 384 billion was spent on clothing and footwear.