College Algebra Word Problem

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)= −4.275 x squared+72.55x+97.24, where X=0 represents 2000, x=1 represents 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
nothing, $
nothing billion was spent on clothing and footwear.
(Round the final answer to the nearest whole number needed. Round all intermediate values to the nearest tenth as needed.)

Answer 1

 

The Quadratic function:

f(x)= −4.275x² +72.55x + 97.24

is a downward facing parabola, you can tell because the highest order coefficient (-4.275) is negative.

 

This means that the maximum will be the vertex.

Standard form:

ax² + bx + c = 0

Vertex: (-b/2a, y)

b = 72.55

a = -4.275

-b/2a = -(72.55)/(2*-4.275) = 8.49

Plug this into the equation and solve for y

f(x) = -4.275(8.49)² + 72.55(8.49) + 97.24

f(8.49) = 405.05

Vertex: (8.49, 405.05)

 

The word problem states that x = number of years from 2000 and y, or f(x) is the amount of money spent. So the maximum amount spent was $405.05 ; 8.49 years after 2000 would be ~ 2008.5

 

To find the year when no money was spent, set f(x) = 0 and solve for x

 

0 = -4.275x² + 72.55x + 97.24

 

Use the quadratic equation:

x = [-b ± √(b² - 4ac)]/2a

 

x = [-72.55 + √(5263.5 + 1662.8)]/-8.55

x = (-72.55 + 83.22)/ -8.55

x = -1.25 years

 

2000 - 1.25 years = 1998.75

 

x = [-72.55 - √(5263.5 + 1662.8)]/-8.55
x = (-72.55 - 83.22)/ -8.55
x = 18.22 years

 

2000 + 18.22 years = 2018.22