QuestionThe population of Austin was 494,000 and grows by $3\%$ yearly. Find $y$ after 10 years using $y=494(1.03)^{t}$ .

Studdy Solution

STEP 1

Assumptions1. The initial population of Austin, Texas, is494,000. . The population increases by3% each year.
3. The growth is exponential, meaning the population increases by a fixed percentage each year.
4. We are looking for the population10 years after the beginning of the decade.

STEP 2

The general form of an exponential growth model is $y = a(1 + r)^t$ , where- $y$ is the final amount- $a$ is the initial amount- $r$ is the growth rate (expressed as a decimal) - $t$ is the time (in years)

STEP 3

In this case, the initial amount $a$ is the initial population, which is494,000. The growth rate $r$ is3%, or0.03 when expressed as a decimal. We can substitute these values into the exponential growth model.
$y =494(1 +0.03)^t$

STEP 4

implify the equation.
$y =494(1.03)^t$ This is the exponential growth model that represents the population of Austin, Texas.

STEP 5

To find the population10 years after the beginning of the decade, we substitute $t =10$ into the equation.
$y =494(1.03)^{10}$

STEP 6

Calculate the value of $y$ .
$y =494(1.03)^{10} \approx664$ This means that the population of Austin, Texas,10 years after the beginning of the decade is approximately664,000.

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QuestionThe population of Austin was 494,000 and grows by $3\%$ yearly. Find $y$ after 10 years using $y=494(1.03)^{t}$ .

Studdy Solution

STEP 1

Assumptions1. The initial population of Austin, Texas, is494,000. . The population increases by3% each year.
3. The growth is exponential, meaning the population increases by a fixed percentage each year.
4. We are looking for the population10 years after the beginning of the decade.

STEP 2

The general form of an exponential growth model is $y = a(1 + r)^t$ , where- $y$ is the final amount- $a$ is the initial amount- $r$ is the growth rate (expressed as a decimal) - $t$ is the time (in years)

STEP 3

In this case, the initial amount $a$ is the initial population, which is494,000. The growth rate $r$ is3%, or0.03 when expressed as a decimal. We can substitute these values into the exponential growth model.
$y =494(1 +0.03)^t$

STEP 4

implify the equation.
$y =494(1.03)^t$ This is the exponential growth model that represents the population of Austin, Texas.

STEP 5

To find the population10 years after the beginning of the decade, we substitute $t =10$ into the equation.
$y =494(1.03)^{10}$

STEP 6

Calculate the value of $y$ .
$y =494(1.03)^{10} \approx664$ This means that the population of Austin, Texas,10 years after the beginning of the decade is approximately664,000.

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QuestionThe population of Austin was 494,000 and grows by 3%3\%3% yearly. Find yyy after 10 years using y=494(1.03)ty=494(1.03)^{t}y=494(1.03)t.

Studdy Solution

STEP 1

Assumptions1. The initial population of Austin, Texas, is494,000. . The population increases by3% each year.3. The growth is exponential, meaning the population increases by a fixed percentage each year.4. We are looking for the population10 years after the beginning of the decade.

STEP 2

The general form of an exponential growth model is y=a(1+r)ty = a(1 + r)^ty=a(1+r)t, where- yyy is the final amount- aaa is the initial amount- rrr is the growth rate (expressed as a decimal) - ttt is the time (in years)

STEP 3

In this case, the initial amount aaa is the initial population, which is494,000. The growth rate rrr is3%, or0.03 when expressed as a decimal. We can substitute these values into the exponential growth model.y=494(1+0.03)ty =494(1 +0.03)^ty=494(1+0.03)t

STEP 4

implify the equation.y=494(1.03)ty =494(1.03)^ty=494(1.03)tThis is the exponential growth model that represents the population of Austin, Texas.

STEP 5

To find the population10 years after the beginning of the decade, we substitute t=10t =10t=10 into the equation.y=494(1.03)10y =494(1.03)^{10}y=494(1.03)10

STEP 6

Calculate the value of yyy.y=494(1.03)10≈664y =494(1.03)^{10} \approx664y=494(1.03)10≈664This means that the population of Austin, Texas,10 years after the beginning of the decade is approximately664,000.

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Studdy solves anything!

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionThe population of Austin was 494,000 and grows by $3\%$ yearly. Find $y$ after 10 years using $y=494(1.03)^{t}$ .

Assumptions1. The initial population of Austin, Texas, is494,000. . The population increases by3% each year.
3. The growth is exponential, meaning the population increases by a fixed percentage each year.
4. We are looking for the population10 years after the beginning of the decade.

The general form of an exponential growth model is $y = a(1 + r)^t$ , where- $y$ is the final amount- $a$ is the initial amount- $r$ is the growth rate (expressed as a decimal) - $t$ is the time (in years)

In this case, the initial amount $a$ is the initial population, which is494,000. The growth rate $r$ is3%, or0.03 when expressed as a decimal. We can substitute these values into the exponential growth model.
$y =494(1 +0.03)^t$

implify the equation.
$y =494(1.03)^t$ This is the exponential growth model that represents the population of Austin, Texas.

To find the population10 years after the beginning of the decade, we substitute $t =10$ into the equation.
$y =494(1.03)^{10}$

Calculate the value of $y$ .
$y =494(1.03)^{10} \approx664$ This means that the population of Austin, Texas,10 years after the beginning of the decade is approximately664,000.