Let C(t) be the total value of US currency (coins and banknotes) in circulation at time. The table gives values of this function from 1980 to 2000, as of September 30, in billions of dollars. Estimate the value of C(1990) .
t 1980 1985 1990 1995 2000
C(t) 129.9 187.3 271.9 409.3 568.6
its asking for C'(1990)
I see that C(1990) is given as 271.9
What are you asking?
I have this question in my homework .its a multiple choice,
Select the correct answer. Answers are in billions of dollars per year.
and the answers are
a)137.4
b)44.4
c)22.2
d)16.96
e)27.48
To estimate the value of C(1990), we can use linear interpolation since we have the values of C(t) for the years 1980, 1985, 1990, 1995, and 2000.
Linear interpolation involves finding the equation of a straight line based on two known points and using it to estimate the value at a specific point between those two known points.
The known points in this case are (1985, 187.3) and (1995, 409.3), which represent the values of C(t) in the years 1985 and 1995, respectively.
To find the equation of the straight line, we need to calculate the slope and the y-intercept.
Slope:
The slope of the line passing through two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)
Using the values:
(x1, y1) = (1985, 187.3)
(x2, y2) = (1995, 409.3)
m = (409.3 - 187.3) / (1995 - 1985)
m = 222 / 10
m = 22.2
Y-intercept:
The y-intercept of the line passing through two points can be found using the formula:
b = y1 - m * x1
Using the point (1985, 187.3):
b = 187.3 - 22.2 * 1985
b = 187.3 - 44037
b ≈ -43849.7
Now that we have the slope (m) and y-intercept (b), we can find the equation of the line:
y = mx + b
Substituting the values:
y = 22.2x - 43849.7
To estimate C(1990), we substitute x = 1990 into the equation:
C(1990) = 22.2 * 1990 - 43849.7
Calculating this gives:
C(1990) ≈ 271.9 billion dollars
Therefore, the estimated value of C(1990) is approximately 271.9 billion dollars.