Suppose the average growth rate for a small town with a population of 35,230 in the northeast United States was approximately 0.65% per year.
(a)
Assuming that growth rate continues, write an equation that will give the population A in t years.
A(t) = 35,230 * (1 + 0.0065)^t
In how many years will the population be 40,000? Round to the nearest tenth of a year.
To find the number of years it will take for the population to reach 40,000, we need to solve the equation:
40,000 = 35,230 * (1 + 0.0065)^t
Dividing both sides by 35,230:
1.1344 = (1.0065)^t
Taking the natural logarithm of both sides to solve for t:
ln(1.1344) = ln(1.0065)^t
ln(1.1344) = t * ln(1.0065)
t = ln(1.1344) / ln(1.0065)
Calculating t:
t ≈ ln(1.1344) / ln(1.0065) ≈ 14.8 years
Therefore, it will take approximately 14.8 years for the population to reach 40,000.