Suppose the price of a first-class stamp was 4¢ for the first time in 1958 and 44¢ in 2009. Find a simple exponential function of the form
y = ab^t that models the cost of a first-class stamp for 1958–2009. (Let
t = 0 correspond to 1958. Assume y is in dollars. Round your value for b to four decimal places.)
Also how to predict value for 2020
there are 51 years from 1958 to 2009
So, you want b such that
4b^51 = 44/4
b^51 = 11/4
b = (11/4)^(1/51) = 1.02003
To find the exponential function that models the cost of a first-class stamp from 1958 to 2009, we need to find the values of a and b in the equation y = ab^t. Let's start by finding the value of b.
Since t = 0 corresponds to 1958, we can use the given data points (1958, 4¢) and (2009, 44¢) to set up two equations:
4 = ab^0 (equation 1, for 1958)
44 = ab^(2009-1958) (equation 2, for 2009)
Simplifying equation 1, we get:
4 = a
Now, let's substitute this value of a into equation 2:
44 = 4b^(51)
Dividing both sides by 4, we have:
11 = b^(51)
To find b, we need to take the 51st root of both sides of the equation:
b = 11^(1/51)
Calculating this value, we get:
b ≈ 1.0867 (rounded to four decimal places)
Now that we have the value of b, we can find the value of a by substituting it into equation 1:
4 = a
Thus, a = 4.
The exponential function that models the cost of a first-class stamp from 1958 to 2009 is:
y = 4 * (1.0867)^t
To predict the cost of a first-class stamp in 2020, we can substitute t = 2020 - 1958 = 62 into the equation:
y = 4 * (1.0867)^62
Calculating this value will give us the predicted cost of a first-class stamp in 2020.