Question

The cost of first-class postage stamp was 7∈/ in 1964 and 70∈/ in 2010. This increase represents exponential growth. Write the function S for the cost of a stamp t years after 1964(t=0). b) What was the growth rate in the cost?

Answer 1

The exponential growth function for the cost of a postage stamp, S(t), years after 1964 is S(t) = 0.07 × (1.0583)^t with an annual growth rate of approximately 5.83%.

Step-by-step explanation:

The student is asking for the function representing the exponential growth in the cost of a first-class postage stamp from 7 cents in 1964 to 70 cents in 2010, which corresponds to t=0 and t=46 respectively. To formulate this function, S(t), we use the general form of an exponential growth equation, which is S(t) = S_0 × (1 + r)^t, where S_0 is the initial value, r is the growth rate, and t is the time elapsed. Substituting the initial value S_0 as 0.07 dollars, and acknowledging that the final value is 0.70 dollars when t equals 46 years, we can set up the equation 0.70 = 0.07 × (1 + r)^{46} to solve for r.

To find the growth rate r, we rearrange and solve for r:

r = ((0.70/0.07)^{1/46}) - 1

Calculating this gives us:

r = (10^{1/46}) - 1 ≈ 0.0583 or 5.83%

Thus, the function for the cost of a stamp t years after 1964 can be written as:

S(t) = 0.07 × (1.0583)^t

And the annual exponential growth rate in the cost is approximately 5.83%.