Question

The population of a town grows exponentially. After 1 year, the population is 34,560. After 2 years, the population is 37,325. Provide an equation that can be used to predict, y, the number of people living in the town after x years. (Round population values to the nearest whole number.)

Answer 1

Equation will be
`y=(32083).(1.08)^(x)`

Explanation:

Since population of a town grows exponentially so the function to represent the population growth will be

`a_(n)=a_(0)(r)^(n)`

n = number of years

where
`a_(n)` = population after n years

`a_(0)` = population in the base year or population at 0 year

r = common ratio or the growth with which population is increasing year by year

Now It is given in the question after 1 year, the population is 34560

So the expression will be
`34560=a_(0)(r)^(1)=a_(0).(r)`------(1)

again question states that after 2 years, the population is 37325

The expression becomes
`a_(2)=37325=a_(0).(r)^(2)`------(2)

Now we divide equation 2 from equation 1

`(37325)/(34560)=(a_(0).r^(2))/(a_(0).r)`

r = 1.08

Now we put the value of "r" in equation 1 to calculate the value of
`a_(0)`

`34650=a_(0).(1.08)`

`a_(0)= (34650)/(1.08)=32083`

If y be the number of people living in the town after x years then the function formed will be

`y=(32083).(1.08)^(x)`

Answer 2

For this case we have a function of the form:

`y = A * (b) ^ x`
Where,
A: initial amount
b: growth rate
x: time in years
We must find the values of A and b.
For this, we use the following data:
After 1 year, the population is 34,560:

`A * (b) ^ 1 = 34560`
After 2 years, the population is 37,325:

`A * (b) ^ 2 = 37325`
We have a system of two equations with two unknowns.
By dividing both equations we obtain the value of b:

`(A * (b) ^ 2)/(A * (b) ^ 1) = (37325)/(34560)`

`b = (37325)/(34560)`

`b = 1.08`
Then, the value of A is obtained by substituting b in any of the equations.
We have then:

`A * (b) ^ 1 = 34560 A * (1.08) ^ 1 = 34560 A * (1.08)= 34560`

`A = (34560)/(1.08)`

`A = 32000`
Answer:
an equation that can be used to predict, y, the number of people living in the town after x years is:

`y=32000*(1.08)^x`