Math  /  Algebra

QuestionHomework 4.3 - Exponential Functions Question 8 of 8 (1 point) I Question Attempt: 1 of Unlimited Juliana
The population of a country was 933,000 in 2009 with an annual growth rate of 0.07%0.07 \%. (a) Find a mathematical model that relates the population of a country as a function of the number of years after 2009. (b) If the annual rate of increase remains the same, use this model to predict the population of a country in the year 2050. Round to the nearest thousand.
Part 1 of 2 (a) The model is P(t)=P(t)= \square Esp

Studdy Solution

STEP 1

1. The initial population in 2009 is 933,000.
2. The annual growth rate is 0.07%0.07\%.
3. We need to find a mathematical model for the population as a function of time.
4. The function should model exponential growth.

STEP 2

1. Identify the form of the exponential growth model.
2. Determine the growth rate in decimal form.
3. Write the exponential growth function.
4. Use the model to predict the population in 2050.

STEP 3

The general form of an exponential growth model is given by:
P(t)=P0×(1+r)t P(t) = P_0 \times (1 + r)^t
where P(t) P(t) is the population at time t t , P0 P_0 is the initial population, r r is the growth rate, and t t is the time in years after the initial year.

STEP 4

Convert the annual growth rate from a percentage to a decimal:
r=0.07%=0.07100=0.0007 r = 0.07\% = \frac{0.07}{100} = 0.0007

STEP 5

Substitute the initial population and growth rate into the exponential growth model:
P(t)=933,000×(1+0.0007)t P(t) = 933,000 \times (1 + 0.0007)^t
This is the mathematical model relating the population as a function of the number of years after 2009.

STEP 6

To predict the population in 2050, calculate the number of years after 2009:
t=20502009=41 t = 2050 - 2009 = 41
Substitute t=41 t = 41 into the model:
P(41)=933,000×(1+0.0007)41 P(41) = 933,000 \times (1 + 0.0007)^{41}
Calculate the population:
P(41)933,000×(1.0007)41 P(41) \approx 933,000 \times (1.0007)^{41}
P(41)933,000×1.0289 P(41) \approx 933,000 \times 1.0289
P(41)959,000 P(41) \approx 959,000
Thus, the predicted population in 2050, rounded to the nearest thousand, is 959,000.
The mathematical model is:
P(t)=933,000×(1.0007)t P(t) = 933,000 \times (1.0007)^t
The predicted population in 2050 is:
959,000 \boxed{959,000}

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