The table lists the price P of a
first-class postage stamp for selected year’s t.
t 1963 1975 1987 2002 2007
p $.05 $.13 $.25 $.37 $.41
(a) Find the slope of each line segment connecting consecutive data points in the table. Can these data be modeled with a line? Explain.
(b) Does the polynomial 0.835t – 1635 model the
data in the table exactly? Explain
I need to turn this in before 7 tonight, Monday, please help
changes in p(cents): 8 12 12 4
changes in t(years): 12 12 15 5
slope on interval(cents/yr): 3/2 1 5/4 5/4
No single line models the data exactly
oops. I wrote the slope reciprocals (years/cent). Still no single line fits all the data points.
(a) To find the slope of each line segment connecting consecutive data points in the table, we can use the formula for slope, which is given by:
Slope = (change in y) / (change in x)
Let's calculate the slopes for the given data points:
From 1963 to 1975:
Slope = (0.13 - 0.05) / (1975 - 1963) = 0.08 / 12 = 0.0067
From 1975 to 1987:
Slope = (0.25 - 0.13) / (1987 - 1975) = 0.12 / 12 = 0.01
From 1987 to 2002:
Slope = (0.37 - 0.25) / (2002 - 1987) = 0.12 / 15 = 0.008
From 2002 to 2007:
Slope = (0.41 - 0.37) / (2007 - 2002) = 0.04 / 5 = 0.008
Now, let's analyze whether these data can be modeled with a line. If the slopes of the line segments were constant, then the relationship between t and P could be represented by a linear equation. However, as we can see, the slopes are not the same for all segments. Therefore, the data cannot be modeled exactly with a single line.
(b) The polynomial 0.835t - 1635 does not model the data in the table exactly. We can verify this by substituting the given years (t values) into the equation and comparing the results with the actual prices (P values).
Let's calculate the values for the polynomial equation:
For t = 1963:
P = (0.835 * 1963) - 1635 = 1640.905 - 1635 = 5.905
For t = 1975:
P = (0.835 * 1975) - 1635 = 1649.125 - 1635 = 14.125
For t = 1987:
P = (0.835 * 1987) - 1635 = 1659.845 - 1635 = 24.845
For t = 2002:
P = (0.835 * 2002) - 1635 = 1672.67 - 1635 = 37.67
For t = 2007:
P = (0.835 * 2007) - 1635 = 1676.145 - 1635 = 41.145
Comparing these calculated values with the actual prices from the table, we can see that they do not match. Therefore, the polynomial 0.835t - 1635 does not model the data exactly.