QuestionModel Clover Hills' population with $y=a(b)^{x}$ , given it grew from 3,650 to 3,869 in 1 year. Predict after 4 years.

Studdy Solution

STEP 1

Assumptions1. The initial population of Clover Hills when Richard moved there was3,650. . The population of Clover Hills one year after Richard moved there is3,869.
3. The population of Clover Hills increases exponentially each year.
4. We are looking for an exponential equation in the form $y=a(b)^{x}$ , where $y$ is the population, $x$ is the number of years after Richard moved there, and $a$ and $b$ are constants we need to find.

STEP 2

First, we need to find the base of the exponential function, $b$ . We can do this by dividing the population one year after Richard moved there by the initial population.
$b = \frac{Population\, after\,1\, year}{Initial\, population}$

STEP 3

Now, plug in the given values for the initial population and the population after one year to calculate $b$ .
$b = \frac{3,869}{3,650}$

STEP 4

Calculate the value of $b$ .
$b = \frac{3,869}{3,650} =1.06$

STEP 5

Now that we have the value of $b$ , we can write the exponential function. The initial population when Richard moved there,3,650, is the value of $a$ in our function.
$y = a(b)^{x}$ $y =3,650(1.06)^{x}$

STEP 6

Now that we have the exponential equation, we can use it to predict the population4 years after Richard moved there. We do this by substituting4 for $x$ in our equation.
$y =3,650(1.06)^{4}$

STEP 7

Calculate the value of $y$ .
$y =3,650(1.06)^{4} \approx4,385$ To the nearest hundred people, Richard can expect the population to be4,400 people4 years after he moved there.

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QuestionModel Clover Hills' population with $y=a(b)^{x}$ , given it grew from 3,650 to 3,869 in 1 year. Predict after 4 years.

Studdy Solution

STEP 1

STEP 2

First, we need to find the base of the exponential function, $b$ . We can do this by dividing the population one year after Richard moved there by the initial population.
$b = \frac{Population\, after\,1\, year}{Initial\, population}$

STEP 3

Now, plug in the given values for the initial population and the population after one year to calculate $b$ .
$b = \frac{3,869}{3,650}$

STEP 4

Calculate the value of $b$ .
$b = \frac{3,869}{3,650} =1.06$

STEP 5

Now that we have the value of $b$ , we can write the exponential function. The initial population when Richard moved there,3,650, is the value of $a$ in our function.
$y = a(b)^{x}$ $y =3,650(1.06)^{x}$

STEP 6

Now that we have the exponential equation, we can use it to predict the population4 years after Richard moved there. We do this by substituting4 for $x$ in our equation.
$y =3,650(1.06)^{4}$

STEP 7

Calculate the value of $y$ .
$y =3,650(1.06)^{4} \approx4,385$ To the nearest hundred people, Richard can expect the population to be4,400 people4 years after he moved there.

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QuestionModel Clover Hills' population with y=a(b)xy=a(b)^{x}y=a(b)x, given it grew from 3,650 to 3,869 in 1 year. Predict after 4 years.

Studdy Solution

STEP 1

STEP 2

STEP 3

Now, plug in the given values for the initial population and the population after one year to calculate bbb.b=3,8693,650b = \frac{3,869}{3,650}b=3,6503,869​

STEP 4

Calculate the value of bbb.b=3,8693,650=1.06b = \frac{3,869}{3,650} =1.06b=3,6503,869​=1.06

STEP 5

Now that we have the value of bbb, we can write the exponential function. The initial population when Richard moved there,3,650, is the value of aaa in our function.y=a(b)xy = a(b)^{x}y=a(b)xy=3,650(1.06)xy =3,650(1.06)^{x}y=3,650(1.06)x

STEP 6

Now that we have the exponential equation, we can use it to predict the population4 years after Richard moved there. We do this by substituting4 for xxx in our equation.y=3,650(1.06)4y =3,650(1.06)^{4}y=3,650(1.06)4

STEP 7

Calculate the value of yyy.y=3,650(1.06)4≈4,385y =3,650(1.06)^{4} \approx4,385y=3,650(1.06)4≈4,385To the nearest hundred people, Richard can expect the population to be4,400 people4 years after he moved there.

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QuestionModel Clover Hills' population with $y=a(b)^{x}$ , given it grew from 3,650 to 3,869 in 1 year. Predict after 4 years.

Now, plug in the given values for the initial population and the population after one year to calculate $b$ .
$b = \frac{3,869}{3,650}$

Calculate the value of $b$ .
$b = \frac{3,869}{3,650} =1.06$

Now that we have the value of $b$ , we can write the exponential function. The initial population when Richard moved there,3,650, is the value of $a$ in our function.
$y = a(b)^{x}$ $y =3,650(1.06)^{x}$

Now that we have the exponential equation, we can use it to predict the population4 years after Richard moved there. We do this by substituting4 for $x$ in our equation.
$y =3,650(1.06)^{4}$

Calculate the value of $y$ .
$y =3,650(1.06)^{4} \approx4,385$ To the nearest hundred people, Richard can expect the population to be4,400 people4 years after he moved there.