Answer :
To solve the problem of predicting the population in year 30, given by the regression equation \( y = 18,000 \left(1.04^x\right) \), we will follow these steps:
1. Understand the given regression equation:
\( y = 18,000 \left(1.04^x\right) \)
- \( y \) represents the population after \( x \) years.
- 18,000 is the initial population.
- 1.04 is the growth rate of the population per year.
- \( x \) is the number of years for which we want to predict the population.
2. Substitute the value of \( x \) with 30:
- We want to find the population after 30 years, so we substitute \( x = 30 \) into the equation.
3. Calculate the population:
- Plugging in 30 for \( x \), the equation becomes:
[tex]\[ y = 18,000 \left(1.04^{30}\right) \][/tex]
4. Evaluate the expression:
- Following the calculation, we find that:
[tex]\[ y = 18,000 \times 1.04^{30} \][/tex]
- Using the exponential growth factor \( 1.04^{30} \):
[tex]\[ 1.04^{30} \approx 3.243397 \][/tex]
5. Multiply the initial population by the growth factor:
- Now, multiply 18,000 by the growth factor:
[tex]\[ y = 18,000 \times 3.243397 \][/tex]
6. Result:
- Performing the multiplication, we get:
[tex]\[ y \approx 58,381.15518049574 \][/tex]
7. Choose the best prediction:
- Comparing with the provided options, the closest and best prediction for the population in year 30 is:
- D. 58,381
Therefore, the best prediction for the population in year 30 is:
D. 58,381.
1. Understand the given regression equation:
\( y = 18,000 \left(1.04^x\right) \)
- \( y \) represents the population after \( x \) years.
- 18,000 is the initial population.
- 1.04 is the growth rate of the population per year.
- \( x \) is the number of years for which we want to predict the population.
2. Substitute the value of \( x \) with 30:
- We want to find the population after 30 years, so we substitute \( x = 30 \) into the equation.
3. Calculate the population:
- Plugging in 30 for \( x \), the equation becomes:
[tex]\[ y = 18,000 \left(1.04^{30}\right) \][/tex]
4. Evaluate the expression:
- Following the calculation, we find that:
[tex]\[ y = 18,000 \times 1.04^{30} \][/tex]
- Using the exponential growth factor \( 1.04^{30} \):
[tex]\[ 1.04^{30} \approx 3.243397 \][/tex]
5. Multiply the initial population by the growth factor:
- Now, multiply 18,000 by the growth factor:
[tex]\[ y = 18,000 \times 3.243397 \][/tex]
6. Result:
- Performing the multiplication, we get:
[tex]\[ y \approx 58,381.15518049574 \][/tex]
7. Choose the best prediction:
- Comparing with the provided options, the closest and best prediction for the population in year 30 is:
- D. 58,381
Therefore, the best prediction for the population in year 30 is:
D. 58,381.
