Answered

A town has a population of 78,918 residents. The town council is offering a prize for the best prediction of how long it will take the population to reach 100,000. The population rate is increasing 6% per year. Find the best prediction in order to win the prize. Write an exponential equation in the form y=ab^x and explain what a and b represents



Answer :

sqdancefan

Answer:

  • y = 78918·1.06^x
  • about 4.063 years

Step-by-step explanation:

You want the time it take for a population of 78,918 to grow to 100,000 at the rate of 6% per year.

Equation

The population can be described by ...

  y = ab^x

where 'a' is the initial population, and 'b' is the growth factor. The growth factor is ...

  b = 1 + growth rate = 1 + 6% = 1.06

The initial population is given as 78,918, so the equation for the population in x years is ...

  y = 78918·1.06^x . . . . . population in x years

Time

Solving for x when y = 100,000 gives ...

  100000 = 78918·1.06^x . . . . . . use desired y value

  100000/78918 = 1.06^x

  log(100000/78918) = x·log(1.06)

  x = log(100000/78918)/log(1.06) = -log(0.78918)/log(1.06) ≈ 4.063

The best prediction of the time it will take for the population to reach 100,000 is 4.063 years.

__

Additional comment

That's about 4 years and 23 days.

View image sqdancefan

Other Questions