Respuesta :
Answer:
The annual increase was approximately $0.002
Explanation:
In order to know the annual compound increase in the cost of the first-class postage during the 54 year period, we need to know the rate at which the compound interest was calculated. We can know that using the following compound interest formula:
[tex]A = P(1 + \frac{r}{100})^{t}[/tex]
P = principal amount (the initial amount for the envelope)
r = annual rate of increase
t = number of years the amount is increased.
A = amount of money accumulated after n years, including the increase.
Now, we have our Amount at the 54th year to be 0.45dollars, when the principal is 0.04dollars.
Therefore, we have
A = $0.45
P = $0.04
r = unknown (that's what we are looking for)
t = 54
Substituting these into the formula, we have:
[tex]0.45 = 0.04(1 + \frac{r}{100})^{54}[/tex]
Dividing both sides by 0.04 we have:
[tex]11.25 = (1 + \frac{r}{100})^{54}[/tex]
Taking the 54th root of both sides we have(approximately):
[tex]1.05 = (1 + \frac{r}{100})[/tex]
The above gives:
[tex]0.05 = \frac{r}{100}[/tex]
This gives:
[tex]r = 5%[/tex]
Therefore, the money increased annually at the rate of 5% approximately, and that would be
[tex]\frac{5}{100} \times 0.04[/tex]
Which is $0.002 approximately.
