Question

WILL GIVE BRAINLIEST AND A LOT OF POINTS!!!! PLEASE A cities population is estimated to be 56,000, and is growing at a yearly rate of 2%. After approximately how many years will the population of the city reach 80,000? Make sure to show all your work.

Answer 1

The population starts at 56,000.
After 1 year, it's (56,000) x (1.02) .
After 2 years, it's (56,000) x (1.02) x (1.02)
After 3 years, it's (56,000) x (1.02) x (1.02) x (1.02)
.
.
After ' n ' years, it's (56,000) x (1.02)ⁿ .

We want to know what ' n ' is when the population reaches 80,000 .

At that time . . .

80,000 = (56,000) x (1.02)ⁿ

Divide each side by 56,000 :

(80/56) = (1.02)ⁿ

Sorry, but now I have to take the log of each side:

log(80) - log(56) = n · log(1.02)

Divide each side by log(1.02) , and you have an expression for ' n ' :

n = [ log(80) - log(56) ] / log(1.02)

n = (1.903 - 1.748) / (0.0086)

n = 18 years

You can check this.
Just stuff (56,000) · (1.02)¹⁸ into your calculator.
I did that, and I got 79,981.8 , which is really awfully close to 80,000 .