Quadratic Formula 2 Part Question

Question

The percent of U.S. households with high-speed internet h can be estimated by: h = -0.2n2 + 7.2n + 1.5 Where n is the number of years since 1990. A. Use the Quadratic Formula to determine when 20% of the population will have high-speed internet. B. Is a quadratic equation a good model for this information? Explain.

Answer 1

h = -.2n²+7.2n+1.5
This will graph as a parabola opening downward.
Thus, there will be a maximum percentage at the vertex
with a decrease in percentage past the vertex.
This means the percentage of homes with high
speed internet will rise for a certain number of years
and then begin to decline back to zero...
 
Given todays world, I do not believe this is a good model
for the information as high speed internet access will
most likely continue to rise to near 100% with little
significant decrease in the foreseeable future.
 
Find the vertex:  v = (-b/2a, f(-b/2a))
-b/2a = -7.2/(2(-.2) = -7.2/(-.4) = 18
The h value at n=18 is
-.2(18²) + 7.2(18) + 1.5 = 66.3%
Thus, your quadratic model shows a maximum percentage
of 66.3% in the year 2008

Getting back to the main problem of when the value will

reach 20%...  Note, this will occur twice on the graph.

Once to the left of the vertex with a value less than 18

years, and once to the right of the vertex with a value

greater than 18 years.

 

h = -.2n²+7.2n+1.5

 

-.2n²+7.2n+1.5=20

-.2n²+7.2n-18.5=0

Let's multiply through by 10 to eliminate the decimals

-2n² + 72n -185 = 0

For quadratic equation a=-2, b=72, c=-185

 

n = [-72 ±√(72²-4(-2)(-185))]/2(-2)

 

n =(-72 ±√3704)/-4

 

n = (-72 ± 60.8605)/-4

 

n = (-72 - 60.8605)/-4 = -132.8605/-4 = 33.2 years....

OR

n = (-72 + 60.8605)/-4 = -11.1395/-4 = 2.8 years

 

Rounding our answers we have n=3 and n=33

 

Thus the percentage of homes with high speed internet service

will be at 20% in 1993 and again in 2023