To find the gross annual income in Kenneth's fifth year of work, you correctly used the formula A = p(1 + i)^n.
A = 3200(1 + 0.07)^5 = 3200(1.07)^5 ≈ 4488.17
So, the annual gross income in Kenneth's fifth year of work would be approximately $4,488.17.
Now, moving on to part b, to find the minimum number of years Kenneth will have to work to earn a gross annual income of at least $60,000, we can rearrange the formula to solve for n.
Starting with:
60,000 = 3200(1 + 0.07)^n
Divide both sides of the equation by 3200:
60,000 / 3200 = (1.07)^n
Simplify the left side:
18.75 = (1.07)^n
To isolate n, take the logarithm of both sides using the base 1.07 (since it's the base in the equation):
log(18.75) = log((1.07)^n)
Using the logarithm property log(a^b) = b * log(a):
log(18.75) = n * log(1.07)
Now, divide both sides by log(1.07):
n = log(18.75) / log(1.07)
Using a calculator or computer program to evaluate this expression:
n ≈ 23.02
So, Kenneth will need to work for at least 23 years to earn a gross annual income of at least $60,000.