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)= -4.273x² +72.58x + 96.69, 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.

Answer 1

Solution:

$\text{we have,}$

$f(x)=-4.273x^2+72.58x+96.69$

$\text{for f to have maximum}\hspace{0.1in}f'(x)=0$

$=>f'(x)=-8.546x+72.58=0$

$=>x=\frac{72.58}{8.546}=8.493$

$\text{To ensure that at the above value of x, f has maximum}\hspace{0.1in} f'' < 0$

$=>f''(x)=-8.546<0$

$\text{Hence f has maximum at}\hspace{0.1in}x=8.493\hspace{0.1in}\text{means the year is 2008 and that maximum value is }$$f_{max}=404.896$