Is my answer correct? The population of a city is 160,000 and is increasing at a rate of 1.5% each year. Approximately when will the population reach 320,000? My answer is approximately 46 years. Is that right? Thx
320,000 = 160,000 (1.015)^n
2 = 1.015^n
log 2 = n log (1.015)
n = .301/.00647 = 46.5 years
To check if your answer is correct, we can use the formula for compound interest:
A = P * (1 + r)^t
Where:
- A is the final population
- P is the initial population
- r is the growth rate per period (expressed as a decimal)
- t is the number of periods
In this case, the initial population (P) is 160,000, the growth rate per year (r) is 1.5% or 0.015, and we want to find the number of years (t) when the population reaches 320,000.
Let's plug in the values and solve for t:
320,000 = 160,000 * (1 + 0.015)^t
Dividing both sides by 160,000:
2 = (1 + 0.015)^t
To solve for t, we can take the logarithm of both sides:
log(2) = log((1 + 0.015)^t)
Using the logarithmic property, we can bring down the exponent:
log(2) = t * log(1.015)
Now, we can isolate t:
t = log(2) / log(1.015)
Calculating this expression, we find that t ≈ 45.98 years.
So, your answer of approximately 46 years is indeed correct. The population will reach 320,000 after around 46 years.