Questions: a) The cost of first-class postage stamp was 6 ¢ in 1962 and 54 ¢ in 2010. This increase represents exponential growth. Write the function S for the cost of a stamp t years after 1962 (t=0). b) What was the growth rate in the cost? c) Predict the cost of a first-class postage stamp in 2013, 2016, and 2019. d) The Forever Stamp is always valid as first-class postage on standard envelopes weighing 1 ounce or less, regardless of any subsequent increases in the first-class rate. An advertising firm spent 5400 on 10,000 first-class postage stamps in 2009. Knowing it will need 10,000 first-class stamps in each of the years 2010-2020, it decides at the beginning of 2010 to try to save money by spending 5400 on 10,000 Forever Stamps, but also buying enough of the stamps to cover the years 2011 through 2020. Assuming there is a postage increase in each of the years 2013, 2016, and 2019 to the cost predicted in part (c), how much money will the firm save by buying the stamps? a) Choose the correct answer below. A. S(t)=6 e^(0.05 t) B. S(t)=0.05 e^(54 t) C. S(t)=0.05 e^(6 t) D. S(t)=6 e^(k t)

a) Write the function \( S(t) \) for the cost of a stamp \( t \) years after 1962.

Determine the growth rate \( k \).

Using the formula for exponential growth, we have:
\[ k = \frac{\ln\left(\frac{S_{\text{final}}}{S_{\text{initial}}}\right)}{t} = \frac{\ln\left(\frac{0.54}{0.06}\right)}{48} \approx 0.04577551202783791 \]

Write the function \( S(t) \).

The function for the cost of a stamp is given by:
\[ S(t) = 0.06 e^{0.04577551202783791 t} \]

\(\boxed{S(t) = 0.06 e^{0.04577551202783791 t}}\)

b) What was the growth rate in the cost?

Identify the growth rate \( k \).

The growth rate calculated is:
\[ k \approx 0.04577551202783791 \]

\(\boxed{k \approx 0.0458}\)

c) Predict the cost of a first-class postage stamp in 2013, 2016, and 2019.

Calculate \( S(2013 - 1962) \).

The cost in 2013 is:
\[ S(51) \approx 0.6194894528375338 \]

Calculate \( S(2016 - 1962) \).

The cost in 2016 is:
\[ S(54) \approx 0.7106799669943462 \]

Calculate \( S(2019 - 1962) \).

The cost in 2019 is:
\[ S(57) \approx 0.8152939701776368 \]

\(\boxed{S(2013) \approx 0.6195, S(2016) \approx 0.7107, S(2019) \approx 0.8153}\)

d) Calculate the savings by buying Forever Stamps.

Calculate the total cost with increases from 2010 to 2020.

The total cost with increases is:
\[ \text{Total Cost} \approx 75461.79819848442 \]

Calculate the savings.

The savings from buying Forever Stamps is:
\[ \text{Savings} \approx 75461.79819848442 - 5400 = 70061.79819848442 \]

\(\boxed{\text{Savings} \approx 70061.80}\)

\(\boxed{S(t) = 0.06 e^{0.04577551202783791 t}}\)
\(\boxed{k \approx 0.0458}\)
\(\boxed{S(2013) \approx 0.6195, S(2016) \approx 0.7107, S(2019) \approx 0.8153}\)
\(\boxed{\text{Savings} \approx 70061.80}\)