Dental services. The national cost C in billions of dollars for dental services can be modeled by the linear equation C=2.85n+30.52, where n is the number of years since 1900 (Health Care Financing Administration,.a) find and interpret the C-intercept for the line.b)find and interpret the n-intercept for the line.c)graph the line for n ranging from 0 through 20.
To find and interpret the C-intercept (y-intercept) and the n-intercept (x-intercept) for the given linear equation, we will substitute the values of C and n into the equation and solve for the intercepts.
a) The C-intercept represents the value of C when n is 0. To find the C-intercept, we substitute n = 0 into the equation C = 2.85n + 30.52:
C = 2.85(0) + 30.52
C = 0 + 30.52
C = 30.52
Interpretation: The C-intercept is 30.52 billion dollars. This means that in the year 1900 (the starting point, as n represents the number of years since 1900), the national cost for dental services was estimated to be 30.52 billion dollars.
b) The n-intercept represents the value of n when C is 0. To find the n-intercept, we substitute C = 0 into the equation C = 2.85n + 30.52 and solve for n:
0 = 2.85n + 30.52
-2.85n = 30.52
n = -30.52 / 2.85
n ≈ -10.70
Interpretation: The n-intercept is approximately -10.70. This means that the linear model predicts that the number of years since 1900 reaches 0 (or goes negative) when the national cost for dental services is 0. However, the interpretation of a negative year does not have any practical meaning in this context.
c) To graph the line, we will plot points for n ranging from 0 to 20 and draw a line connecting them.
n | C
-------
0 | 30.52
1 | 2.85(1) + 30.52
2 | 2.85(2) + 30.52
...
20 | 2.85(20) + 30.52
Calculating the values of C for each corresponding n, we get:
n | C
-------
0 | 30.52
1 | 33.37
2 | 36.22
...
20 | 81.52
Plotting these points on a graph with n on the x-axis and C on the y-axis, we can then connect the points with a line. The line will have a positive slope of 2.85.
To find the C-intercept for the line, we set n = 0 in the equation C = 2.85n + 30.52:
C = 2.85(0) + 30.52
C = 0 + 30.52
C = 30.52
Therefore, the C-intercept for the line is 30.52 billion dollars. This represents the estimated cost of dental services in the year 1900.
To find the n-intercept for the line, we set C = 0 in the equation C = 2.85n + 30.52:
0 = 2.85n + 30.52
-30.52 = 2.85n
n = -30.52 / 2.85
n ≈ -10.71
Therefore, the n-intercept for the line is approximately -10.71. This represents the estimated year when the cost of dental services was zero. Keep in mind that since n represents the number of years since 1900, negative values indicate years before 1900.
To graph the line for n ranging from 0 through 20, we'll plot several points and connect them:
n = 0:
C = 2.85(0) + 30.52 = 30.52
(0, 30.52)
n = 5:
C = 2.85(5) + 30.52 ≈ 43.77
(5, 43.77)
n = 10:
C = 2.85(10) + 30.52 ≈ 57.02
(10, 57.02)
n = 15:
C = 2.85(15) + 30.52 ≈ 70.27
(15, 70.27)
n = 20:
C = 2.85(20) + 30.52 ≈ 83.52
(20, 83.52)
Plotting these points and connecting them with a straight line, you will get the graph.