Thank you for visiting A national census bureau predicts that a certain population will increase from 35 9 million in 2000 to 63 7 million in 2050 Complete parts. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
Sure! Let's find the exponential function that models the predicted population growth.
We are given that the population is 35.9 million in the year 2000 and is predicted to be 63.7 million in 2050. We need to find an exponential function of the form [tex]\( f(t) = y_0 \cdot b^t \)[/tex], where [tex]\( y_0 \)[/tex] is the initial population, [tex]\( b \)[/tex] is the growth factor, and [tex]\( t \)[/tex] is the time in years since 2000.
### Step-by-Step Solution:
1. Identify the Initial Population and Future Population:
- The initial population ([tex]\( y_0 \)[/tex]) in the year 2000 is [tex]\( 35.9 \)[/tex] million.
- The population in the year 2050 is projected to be [tex]\( 63.7 \)[/tex] million.
2. Determine the Time Span:
- The time span from the year 2000 to the year 2050 is 50 years. So, [tex]\( t = 50 \)[/tex].
3. Set Up the Exponential Equation:
- You use the formula for exponential growth:
[tex]\[
f(t) = y_0 \cdot b^t
\][/tex]
- Plugging in the known values:
[tex]\[
63.7 = 35.9 \cdot b^{50}
\][/tex]
4. Solve for the Growth Factor [tex]\( b \)[/tex]:
- Divide both sides of the equation by 35.9 to isolate [tex]\( b^{50} \)[/tex]:
[tex]\[
b^{50} = \frac{63.7}{35.9}
\][/tex]
- Compute the right-hand side:
[tex]\[
b^{50} \approx 1.7744
\][/tex]
- Solve for [tex]\( b \)[/tex] by taking the 50th root:
[tex]\[
b \approx (1.7744)^{1/50} \approx 1.0115
\][/tex]
5. Write the Exponential Function:
- Now, substitute [tex]\( y_0 = 35.9 \)[/tex] and [tex]\( b = 1.0115 \)[/tex] into the exponential function form:
[tex]\[
f(t) = 35.9 \cdot 1.0115^t
\][/tex]
So, the exponential function modeling the population growth is [tex]\( f(t) = 35.9 \cdot 1.0115^t \)[/tex]. This function describes the population in millions as a function of the number of years since 2000.
We are given that the population is 35.9 million in the year 2000 and is predicted to be 63.7 million in 2050. We need to find an exponential function of the form [tex]\( f(t) = y_0 \cdot b^t \)[/tex], where [tex]\( y_0 \)[/tex] is the initial population, [tex]\( b \)[/tex] is the growth factor, and [tex]\( t \)[/tex] is the time in years since 2000.
### Step-by-Step Solution:
1. Identify the Initial Population and Future Population:
- The initial population ([tex]\( y_0 \)[/tex]) in the year 2000 is [tex]\( 35.9 \)[/tex] million.
- The population in the year 2050 is projected to be [tex]\( 63.7 \)[/tex] million.
2. Determine the Time Span:
- The time span from the year 2000 to the year 2050 is 50 years. So, [tex]\( t = 50 \)[/tex].
3. Set Up the Exponential Equation:
- You use the formula for exponential growth:
[tex]\[
f(t) = y_0 \cdot b^t
\][/tex]
- Plugging in the known values:
[tex]\[
63.7 = 35.9 \cdot b^{50}
\][/tex]
4. Solve for the Growth Factor [tex]\( b \)[/tex]:
- Divide both sides of the equation by 35.9 to isolate [tex]\( b^{50} \)[/tex]:
[tex]\[
b^{50} = \frac{63.7}{35.9}
\][/tex]
- Compute the right-hand side:
[tex]\[
b^{50} \approx 1.7744
\][/tex]
- Solve for [tex]\( b \)[/tex] by taking the 50th root:
[tex]\[
b \approx (1.7744)^{1/50} \approx 1.0115
\][/tex]
5. Write the Exponential Function:
- Now, substitute [tex]\( y_0 = 35.9 \)[/tex] and [tex]\( b = 1.0115 \)[/tex] into the exponential function form:
[tex]\[
f(t) = 35.9 \cdot 1.0115^t
\][/tex]
So, the exponential function modeling the population growth is [tex]\( f(t) = 35.9 \cdot 1.0115^t \)[/tex]. This function describes the population in millions as a function of the number of years since 2000.
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