Math / Algebra

QuestionPoints: 0 of 1
Assume the carrying copacity of the Earth is 27 billion, Use the 1960 annual growth rate of $2.1 \%$ and population of 3 billion to predict the base growth rate and current growth rate with a logistio model, Assume a current world population of 7.8 billion. How does the prodictod growth rate compare to the actual growth rate of about $1,1 \%$ per year?
What is the base growth rate? $\square$ \% (Round to four docimal places as needed.)

Studdy Solution

STEP 1

What is this asking? Given Earth's carrying capacity, the 1960 population and growth rate, and the current population, we need to find the base growth rate and the current growth rate using a logistic model, and compare the predicted current growth rate to the actual current growth rate of $1.1\%$ . Watch out! Don't mix up the different growth rates: the 1960 rate, the base rate, and the current rate.
Also, remember to convert percentages to decimals for calculations.

STEP 2

1. Find the base growth rate
2. Find the current growth rate
3. Compare the predicted and actual current growth rates

STEP 3

The logistic growth model is given by $\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$ , where $\frac{dP}{dt}$ is the rate of population change, $r$ is the base growth rate, $P$ is the current population, and $K$ is the carrying capacity.

STEP 4

We're given that in 1960, $P = 3$ billion and $\frac{dP}{dt} = 0.021 \cdot 3$ billion per year.
We're also given that $K = 27$ billion.
Let's plug these values into our formula: $0.021 \cdot 3 = r \cdot 3 \left(1 - \frac{3}{27}\right)$ .

STEP 5

We want to isolate $r$ .
First, simplify the equation: $0.063 = 3r\left(\frac{24}{27}\right)$ , which simplifies to $0.063 = r \cdot \frac{72}{27}$ .
Now, divide both sides by $\frac{72}{27}$ (which is the same as multiplying by $\frac{27}{72}$ ): $r = 0.063 \cdot \frac{27}{72} = 0.023625$ .
So, the base growth rate is $\textbf{0.023625}$ .

STEP 6

Now, we'll use the same logistic model formula, but with the current population, $P = 7.8$ billion, and the base growth rate we just found, $r = 0.023625$ .
We still have $K = 27$ billion.

STEP 7

Plugging in the values, we get $\frac{dP}{dt} = 0.023625 \cdot 7.8 \left(1 - \frac{7.8}{27}\right)$ .
This simplifies to $\frac{dP}{dt} = 0.184275 \left(\frac{19.2}{27}\right) \approx 0.13068$ .
So, the predicted current growth rate is approximately $\textbf{0.013068}$ per year, or about $1.3068\%$ .

STEP 8

The predicted current growth rate is about $1.3068\%$ , while the actual current growth rate is given as $1.1\%$ .

STEP 9

The predicted growth rate is higher than the actual growth rate.

STEP 10

The base growth rate is $2.3625\%$ .
The predicted current growth rate using the logistic model is approximately $1.3068\%$ , which is higher than the actual current growth rate of $1.1\%$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the base growth rate
2. Find the current growth rate
3. Compare the predicted and actual current growth rates

STEP 3

The logistic growth model is given by $\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$ , where $\frac{dP}{dt}$ is the rate of population change, $r$ is the base growth rate, $P$ is the current population, and $K$ is the carrying capacity.

STEP 4

We're given that in 1960, $P = 3$ billion and $\frac{dP}{dt} = 0.021 \cdot 3$ billion per year.
We're also given that $K = 27$ billion.
Let's plug these values into our formula: $0.021 \cdot 3 = r \cdot 3 \left(1 - \frac{3}{27}\right)$ .

STEP 5

STEP 6

Now, we'll use the same logistic model formula, but with the current population, $P = 7.8$ billion, and the base growth rate we just found, $r = 0.023625$ .
We still have $K = 27$ billion.

STEP 7

STEP 8

The predicted current growth rate is about $1.3068\%$ , while the actual current growth rate is given as $1.1\%$ .

STEP 9

The predicted growth rate is higher than the actual growth rate.

STEP 10

The base growth rate is $2.3625\%$ .
The predicted current growth rate using the logistic model is approximately $1.3068\%$ , which is higher than the actual current growth rate of $1.1\%$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the base growth rate2. Find the current growth rate3. Compare the predicted and actual current growth rates

STEP 3

The logistic growth model is given by dPdt=rP(1−PK)\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)dtdP​=rP(1−KP​), where dPdt\frac{dP}{dt}dtdP​ is the rate of population change, rrr is the base growth rate, PPP is the current population, and KKK is the carrying capacity.

STEP 4

STEP 5

STEP 6

Now, we'll use the same logistic model formula, but with the current population, P=7.8P = 7.8P=7.8 billion, and the base growth rate we just found, r=0.023625r = 0.023625r=0.023625.We still have K=27K = 27K=27 billion.

STEP 7

STEP 8

The predicted current growth rate is about 1.3068%1.3068\%1.3068%, while the actual current growth rate is given as 1.1%1.1\%1.1%.

STEP 9

The predicted growth rate is higher than the actual growth rate.

STEP 10

The base growth rate is 2.3625%2.3625\%2.3625%.The predicted current growth rate using the logistic model is approximately 1.3068%1.3068\%1.3068%, which is higher than the actual current growth rate of 1.1%1.1\%1.1%.

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Start learning now

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

1. Find the base growth rate
2. Find the current growth rate
3. Compare the predicted and actual current growth rates

The logistic growth model is given by $\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$ , where $\frac{dP}{dt}$ is the rate of population change, $r$ is the base growth rate, $P$ is the current population, and $K$ is the carrying capacity.

Now, we'll use the same logistic model formula, but with the current population, $P = 7.8$ billion, and the base growth rate we just found, $r = 0.023625$ .
We still have $K = 27$ billion.

The predicted current growth rate is about $1.3068\%$ , while the actual current growth rate is given as $1.1\%$ .

The base growth rate is $2.3625\%$ .
The predicted current growth rate using the logistic model is approximately $1.3068\%$ , which is higher than the actual current growth rate of $1.1\%$ .