Question

Devise the exponential growth function that fits the given data, then answer to accompanying question. Be sure to identify the references point (t = 0) and The current population of a town is 70,000 and is growing exponentially. If the population to be 75,000 in 10 years, then what will be the populations 20 years What is the reference point (t = 0)? the initial population 70,000 the current year the population in 10 years, 75,000 What are the units of time? percent people decades years Write the exponential growth function. Round any numerical values to three decimal places as needed. y(t) = What is the population 20 years from now?

Answer 1

In 20 years, the population will be about 80.3 thousand people

Explanation:

If our first time is 0 and the population that goes along with that time is 70,000, we have a coordinate point where x is the time (0), and y is the population at that time (70). Our next time is 10 years later, when the population is 75,000. The coordinate point for that set of data is (10, 75). Now we will use those 2 points in the standard form of an exponential equation to write the model for this particular situation.

Exponential equations are of the form

`y=a(b)^x`

where x and y are the coordinates from our points, one at a time; a is the initial value, and b is the growth rate. Filling in an equation with the first set of data:

`70=a(b)^0`

Anything raised to the power of 0 = 1, so b to the power of 0 = 1 and we simply have that a = 70.

Now we use that value of a along with the x and y from the next coordinate pair to solve for b:

`75=70(b)^(10)`

Begin by dividing both sides by 70 to get

`1.071428571=b^(10)`

Undo the power of 10 on the right by taking the 10th root of both sides:

`(1.071428571)^{(1)/(10)}=(b^(10))^{(1)/(10)}`

On the right side we simply have b now, and on the left we have

1.006923142=b

Now we have a and b to write the model for this situation:

`y=70(1.006923142)^x`

We need to find y, the population, in x = 20 years:

`y=70(1.006923142)^(20)`

Raise the parenthesis to the 20th power giving you

y = 70(1.147959784) and

y = 80.3 thousand people