Struggling with the quadratic formula in this problem.

Question

The population of the United States in 2007-2050 can be modeled by the quadratic function P (t)=0.0022675t^2 + 2.6804t + 300.76 where P represents the population in millions and t is the number of years since 2007. Use the Quadratic formula to solve  P (t)=400 then explain what the solution means.

Answer 1

When P(t) = 400

 

400= 0.0022675t²+2.6804t+300.76

 

0= 0.0022675t2+2.6084t - 99.24                        Subtract 400 from both sides.

 

a = 0.002675                 b = 2.6804                     c = -99.24

 

t =[ - (2.6804) ±√((2.6084)² - 4(0.0022675)(-99.24))]÷2(0.0022675)

 

t = 35.93 or  t = -1218.03

 

Since t represents time since 2007, we can disregard the negative answer.

So after 35.93 years since 2007, the model would predict that there would be 400 million people.  We are looking at 2007 plus 35.93 or the year 2042.93 or about 2043 to reach a population of 400 million.

Answer 2

To use the quadratic formula, we must solve  0.0022675t² + 2.6804 + 300.76 = 400. We need to put the equation in standard form by subtracting 400 from both sides of the equation to get 

                                            0.0022675t² +2.6804t - 99.24 = 0

 

Substituting the values of a, b and c into the quadratic formula results in 

 

                         t = (-2.6804±√(2.6804)² - 4(0.0022675)(-99.24))/ 2(0.0022675)

The radical symbol extends all the way over -99.24

Use a calculator to evaluate the two solutions of 30.45832972 and  -1032.477021.

The two solutions represent the number of years from 2007. The quadratic function models the population from 2007 to 2050. Therefore disregard the negative solution. The positive solution of 30.45832972 signifies that the population will be 400 million in a little over 30 years or approximately 30 years, 5 months and 15 days.

Answer 3

This question is a straight forward application of the quadratic formula, and also an exercise in evaluating answers in context.

 

The underlying question is "How many years after 2007 will the population of the US reach 400 million?"

 

First, the math:

 

We will need a solution for t in the expression

 

0.0022675t^2 + 2.6804t + 300.76 = 400

 

This equation can be rewritten to standard form of 

 

ax²+bx+c = 0

 

0.0022675t^2 + 2.6804t - -99.24 = 0

 

So the values for a, b and c are

 

a = 0.0022675

b = 2.6804

c = -99.24

 

Use the quadratic formula to solve for t

 

t = -b±√(b²-4ac)/2a

 

When you solve that, you will end up with two solutions:

 

t = -1218.03 and t = 35.9321

 

Since we want to know how many years it will take to reach a population of 400 million, we can discard to negative solution.

 

Applying the remaining solution, we can predict that the population of the US will reach 400 million 35.9 years after 2007, or in about 2043.