Math / Algebra

QuestionA population numbers 19,000 organisms initially and grows by 17.4\% each year. Suppose $P$ represents population, and $t$ the number of years of growth. An exponential model for the population can be written in the form $P=a \cdot b^{t}$ where $P=$

Studdy Solution

STEP 1

What is this asking? We need to find a formula that tells us the population at any year, given that it grows by a certain percentage every year. Watch out! Don't mix up the initial population and the growth rate.
Also, remember to convert the percentage to a decimal!

STEP 2

1. Define the function
2. Calculate the result

STEP 3

Alright, so we're given the general form of an exponential function: $P = a \cdot b^{t}$ .
Here, $P$ is the population after $t$ years, $a$ is the initial population, and $b$ is the growth factor.

STEP 4

We're told the initial population is 19,000, so $a = 19,000$ .
Woohoo! One piece down!

STEP 5

Now, the population grows by 17.4% each year.
This means that after one year, the population is 100% + 17.4% = 117.4% of what it was before.
As a decimal, this is 1.174.
So, our growth factor, $b$ , is 1.174.

STEP 6

Putting it all together, our exponential model is $P = 19,000 \cdot (1.174)^{t}$ .
Boom!

STEP 7

The problem asks for the exponential model, which we just found!
It's $P = 19,000 \cdot (1.174)^{t}$ .
We're done!
High five!

STEP 8

$P = 19,000 \cdot (1.174)^{t}$

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QuestionA population numbers 19,000 organisms initially and grows by 17.4\% each year. Suppose $P$ represents population, and $t$ the number of years of growth. An exponential model for the population can be written in the form $P=a \cdot b^{t}$ where $P=$

Studdy Solution

STEP 1

What is this asking? We need to find a formula that tells us the population at any year, given that it grows by a certain percentage every year. Watch out! Don't mix up the initial population and the growth rate.
Also, remember to convert the percentage to a decimal!

STEP 2

1. Define the function
2. Calculate the result

STEP 3

Alright, so we're given the general form of an exponential function: $P = a \cdot b^{t}$ .
Here, $P$ is the population after $t$ years, $a$ is the initial population, and $b$ is the growth factor.

STEP 4

We're told the initial population is 19,000, so $a = 19,000$ .
Woohoo! One piece down!

STEP 5

Now, the population grows by 17.4% each year.
This means that after one year, the population is 100% + 17.4% = 117.4% of what it was before.
As a decimal, this is 1.174.
So, our growth factor, $b$ , is 1.174.

STEP 6

Putting it all together, our exponential model is $P = 19,000 \cdot (1.174)^{t}$ .
Boom!

STEP 7

The problem asks for the exponential model, which we just found!
It's $P = 19,000 \cdot (1.174)^{t}$ .
We're done!
High five!

STEP 8

$P = 19,000 \cdot (1.174)^{t}$

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QuestionA population numbers 19,000 organisms initially and grows by 17.4\% each year. Suppose PPP represents population, and ttt the number of years of growth. An exponential model for the population can be written in the form P=a⋅btP=a \cdot b^{t}P=a⋅bt where P=P=P=

Studdy Solution

STEP 1

What is this asking? We need to find a formula that tells us the population at any year, given that it grows by a certain percentage every year. Watch out! Don't mix up the **initial population** and the **growth rate**.Also, remember to convert the percentage to a decimal!

STEP 2

1. Define the function2. Calculate the result

STEP 3

Alright, so we're given the general form of an exponential function: P=a⋅btP = a \cdot b^{t}P=a⋅bt.Here, PPP is the population after ttt years, aaa is the **initial population**, and bbb is the **growth factor**.

STEP 4

We're told the **initial population** is **19,000**, so a=19,000a = 19,000a=19,000.Woohoo! One piece down!

STEP 5

Now, the population grows by **17.4%** each year.This means that after one year, the population is 100% + 17.4% = 117.4% of what it was before.As a decimal, this is **1.174**.So, our **growth factor**, bbb, is **1.174**.

STEP 6

Putting it all together, our exponential model is P=19,000⋅(1.174)tP = 19,000 \cdot (1.174)^{t}P=19,000⋅(1.174)t.Boom!

STEP 7

The problem asks for the exponential model, which we just found!It's P=19,000⋅(1.174)tP = 19,000 \cdot (1.174)^{t}P=19,000⋅(1.174)t.We're done!High five!

STEP 8

P=19,000⋅(1.174)tP = 19,000 \cdot (1.174)^{t}P=19,000⋅(1.174)t

Get Studdy

Studdy solves anything!

Start learning now

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

QuestionA population numbers 19,000 organisms initially and grows by 17.4\% each year. Suppose $P$ represents population, and $t$ the number of years of growth. An exponential model for the population can be written in the form $P=a \cdot b^{t}$ where $P=$

What is this asking? We need to find a formula that tells us the population at any year, given that it grows by a certain percentage every year. Watch out! Don't mix up the initial population and the growth rate.
Also, remember to convert the percentage to a decimal!

1. Define the function
2. Calculate the result

Alright, so we're given the general form of an exponential function: $P = a \cdot b^{t}$ .
Here, $P$ is the population after $t$ years, $a$ is the initial population, and $b$ is the growth factor.

We're told the initial population is 19,000, so $a = 19,000$ .
Woohoo! One piece down!

Now, the population grows by 17.4% each year.
This means that after one year, the population is 100% + 17.4% = 117.4% of what it was before.
As a decimal, this is 1.174.
So, our growth factor, $b$ , is 1.174.

Putting it all together, our exponential model is $P = 19,000 \cdot (1.174)^{t}$ .
Boom!

The problem asks for the exponential model, which we just found!
It's $P = 19,000 \cdot (1.174)^{t}$ .
We're done!
High five!

$P = 19,000 \cdot (1.174)^{t}$