Between 1919 and 1995, the rate of change in the rate of change of the postage required to mail a first-class 1-ounce letter was approximately 0.022 cent per year squared. The postage was 2 cents in 1919, and it was increasing at the rate of approximately 0.393 cent per year in 1958. Using this information, estimate the postage that will be required to mail a first-class, 1-ounce letter in 2088.
A) $3.46
B) $3.26
C) $2.38
D) $1.46
E) $1.56

Question

Quizplus · Accepted Answer

Let P(t) be the postage required to mail a first-class 1-ounce letter in t years after 1919. Then we are given:

P(0) = 2 cents
P'(39) = 0.393 cent per year
P''(t) = 0.022 cent per year squared

Using the second derivative to approximate P(t), we have:

P(t) ≈ P(39) + P'(39)(t-39) + (1/2)P''(39)(t-39)^2

P(t) ≈ 2 + 0.393(t-39) + 0.011(t-39)^2

To find P(169), we put t = 169 in the above equation and get:

P(169) ≈ 2 + 0.393(169-39) + 0.011(169-39)^2

P(169) ≈ 238 cents = $2.38

Therefore, the postage that will be required to mail a first-class, 1-ounce letter in 2088 is $2.38. So the answer is C.

Between 1919 and 1995, the Rate of Change in the Rate