Math / Data & Statistics

QuestionComplete the table shown to the right for the population growth model for a certain country. \begin{tabular}{|c|c|c|} \hline 2006 Population(millions) & Projected 2030 Population (millions) & Projected Growth Rate, k \\ \hline 77.9 & & 0.0172 \\ \hline \end{tabular} he projected 2030 population is $\square$ million. zound to one decimal place as needed.)

Studdy Solution

STEP 1

What is this asking? Given a country's population in 2006 and its growth rate, what will the population be in 2030? Watch out! Make sure to use the correct time difference between the initial and projected years, and don't mix up the initial and final populations.

STEP 2

1. Define the exponential growth formula
2. Calculate the time difference
3. Calculate the projected population

STEP 3

The exponential growth formula is given by $P(t) = P_0 \cdot e^{kt}$ , where: $P(t)$ is the population at time $t$ . $P_0$ is the initial population. $k$ is the growth rate. $t$ is the time in years.

STEP 4

We're given the initial year, 2006, and the target year, 2030.
The time difference is $2030 - 2006 = \mathbf{24}$ years.
So, our $t$ value is 24.

STEP 5

We know the initial population $P_0 = \mathbf{77.9}$ million, the growth rate $k = \mathbf{0.0172}$ , and the time $t = \mathbf{24}$ years.
Let's plug these values into our formula:

STEP 6

$P(24) = 77.9 \cdot e^{0.0172 \cdot 24}$

STEP 7

First, let's calculate the exponent: $0.0172 \cdot 24 = \mathbf{0.4128}$ .

STEP 8

Now, let's calculate $e$ raised to the power of our result: $e^{0.4128} \approx \mathbf{1.5107}$ .
Remember, $e$ is approximately 2.71828.

STEP 9

Finally, multiply this by the initial population: $77.9 \cdot 1.5107 \approx \mathbf{117.7}$ million.

STEP 10

The projected 2030 population is 117.7 million.

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Studdy Solution

STEP 1

What is this asking? Given a country's population in 2006 and its growth rate, what will the population be in 2030? Watch out! Make sure to use the correct time difference between the initial and projected years, and don't mix up the initial and final populations.

STEP 2

1. Define the exponential growth formula
2. Calculate the time difference
3. Calculate the projected population

STEP 3

The exponential growth formula is given by $P(t) = P_0 \cdot e^{kt}$ , where: $P(t)$ is the population at time $t$ . $P_0$ is the initial population. $k$ is the growth rate. $t$ is the time in years.

STEP 4

We're given the initial year, 2006, and the target year, 2030.
The time difference is $2030 - 2006 = \mathbf{24}$ years.
So, our $t$ value is 24.

STEP 5

We know the initial population $P_0 = \mathbf{77.9}$ million, the growth rate $k = \mathbf{0.0172}$ , and the time $t = \mathbf{24}$ years.
Let's plug these values into our formula:

STEP 6

$P(24) = 77.9 \cdot e^{0.0172 \cdot 24}$

STEP 7

First, let's calculate the exponent: $0.0172 \cdot 24 = \mathbf{0.4128}$ .

STEP 8

Now, let's calculate $e$ raised to the power of our result: $e^{0.4128} \approx \mathbf{1.5107}$ .
Remember, $e$ is approximately 2.71828.

STEP 9

Finally, multiply this by the initial population: $77.9 \cdot 1.5107 \approx \mathbf{117.7}$ million.

STEP 10

The projected 2030 population is 117.7 million.

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Studdy Solution

STEP 1

What is this asking? Given a country's population in 2006 and its growth rate, what will the population be in 2030? Watch out! Make sure to use the correct time difference between the initial and projected years, and don't mix up the initial and final populations.

STEP 2

1. Define the exponential growth formula2. Calculate the time difference3. Calculate the projected population

STEP 3

The **exponential growth formula** is given by P(t)=P0⋅ektP(t) = P_0 \cdot e^{kt}P(t)=P0​⋅ekt, where: P(t)P(t)P(t) is the population at time ttt. P0P_0P0​ is the **initial population**. kkk is the **growth rate**. ttt is the **time** in years.

STEP 4

We're given the **initial year**, 2006, and the **target year**, 2030.The time difference is 2030−2006=242030 - 2006 = \mathbf{24}2030−2006=24 **years**.So, our ttt value is **24**.

STEP 5

We know the **initial population** P0=77.9P_0 = \mathbf{77.9}P0​=77.9 million, the **growth rate** k=0.0172k = \mathbf{0.0172}k=0.0172, and the **time** t=24t = \mathbf{24}t=24 years.Let's plug these values into our formula:

STEP 6

P(24)=77.9⋅e0.0172⋅24P(24) = 77.9 \cdot e^{0.0172 \cdot 24}P(24)=77.9⋅e0.0172⋅24

STEP 7

First, let's calculate the exponent: 0.0172⋅24=0.41280.0172 \cdot 24 = \mathbf{0.4128}0.0172⋅24=0.4128.

STEP 8

Now, let's calculate eee raised to the power of our result: e0.4128≈1.5107e^{0.4128} \approx \mathbf{1.5107}e0.4128≈1.5107.Remember, eee is approximately 2.71828.

STEP 9

Finally, multiply this by the **initial population**: 77.9⋅1.5107≈117.777.9 \cdot 1.5107 \approx \mathbf{117.7}77.9⋅1.5107≈117.7 million.

STEP 10

The projected 2030 population is **117.7** million.

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1. Define the exponential growth formula
2. Calculate the time difference
3. Calculate the projected population

The exponential growth formula is given by $P(t) = P_0 \cdot e^{kt}$ , where: $P(t)$ is the population at time $t$ . $P_0$ is the initial population. $k$ is the growth rate. $t$ is the time in years.

We're given the initial year, 2006, and the target year, 2030.
The time difference is $2030 - 2006 = \mathbf{24}$ years.
So, our $t$ value is 24.

We know the initial population $P_0 = \mathbf{77.9}$ million, the growth rate $k = \mathbf{0.0172}$ , and the time $t = \mathbf{24}$ years.
Let's plug these values into our formula:

$P(24) = 77.9 \cdot e^{0.0172 \cdot 24}$

First, let's calculate the exponent: $0.0172 \cdot 24 = \mathbf{0.4128}$ .

Now, let's calculate $e$ raised to the power of our result: $e^{0.4128} \approx \mathbf{1.5107}$ .
Remember, $e$ is approximately 2.71828.

Finally, multiply this by the initial population: $77.9 \cdot 1.5107 \approx \mathbf{117.7}$ million.

The projected 2030 population is 117.7 million.