Suppose that the following table shows the number of millions of people in the United States who lived below the poverty level for selected years. Find a cubic model that approximately fits the data, using x as the number of years after 1960. Round numerical values in your answer to four decimal places. Depending on the technology you use, your answer may be slightly different than the correct answer shown.

Question

Quizplus · Accepted Answer

To fit a cubic model, we need four coefficients (a, b, c, d). We can set up the following system of equations to solve for these coefficients:

a + b(0) + c(0)^2 + d(0)^3 = 39.5
a + b(5) + c(5)^2 + d(5)^3 = 47.2
a + b(10) + c(10)^2 + d(10)^3 = 52.9
a + b(15) + c(15)^2 + d(15)^3 = 56.9

Simplifying each equation, we get:

a = 39.5
a + 5b + 25c + 125d = 47.2
a + 10b + 100c + 1000d = 52.9
a + 15b + 225c + 3375d = 56.9

Substituting a = 39.5 in the second equation, we get:

5b + 25c + 125d = 7.7

Substituting a = 39.5 in the third equation, we get:

10b + 100c + 1000d = 13.4

Subtracting the second equation from the third equation, we get:

5b + 75c + 875d = 5.7

Substituting a = 39.5 in the fourth equation, we get:

15b + 225c + 3375d = 17.4

Subtracting three times the second equation from the fourth equation, we get:

0b + 0c + 0d = 0

This is expected, since a cubic model has four coefficients, and we have four data points.

Now we can solve the system of equations:

a = 39.5
5b + 25c + 125d = 7.7
10b + 100c + 1000d = 13.4

Substituting d = 0.0002b - 0.009c + 0.425, we get:

a = 39.5
11b - 11c = -25.3

Substituting b = c + 2.3, we get:

a = 39.5
22c + 25.3 = 0

Solving for c, we get:

c = -1.150
b = 1.150
d = 0.282

Substituting these values into d = 0.0002b - 0.009c + 0.425, we get:

d = 0.282

Therefore, the cubic model that approximately fits the data is:

f(x) = 39.5 + 1.150x - 1.150x^2 + 0.282x^3

Suppose That the Following Table Shows the Number of Millions