Question

the relationship between the number of years (x) that have passed and the population count per year
number of years (x) | population (y)
1 | 1,440
2 | 1,728
3 | 2,074
4 | 2,488
5 | 2,986
6 | 3,583
7 | 4,300
8 | 5,160
a regression equation for this data is $y = 1,200(1.2)^x$.
based on this data, which value is the best prediction for the town’s population after 12 years?
a. 6,020
b. 8,600
c. 8,916
d. 10,699

Explanation:

Step1: Identify the regression equation and the value of x

The regression equation is \( y = 1200(1.2)^x \), and we need to find the population after 12 years, so \( x = 12 \).

Step2: Substitute x = 12 into the equation

Substitute \( x = 12 \) into \( y = 1200(1.2)^x \). First, calculate \( (1.2)^{12} \).
We know that \( (1.2)^{12}\approx 8.916 \) (using a calculator for exponentiation: \( 1.2\times1.2 = 1.44 \), \( 1.44\times1.2 = 1.728 \), and so on, or using the formula \( a^n=e^{n\ln a} \), \( \ln(1.2)\approx0.1823 \), \( 12\times0.1823\approx2.1876 \), \( e^{2.1876}\approx8.916 \)).

Step3: Calculate y

Now, multiply by 1200: \( y = 1200\times8.916 = 10699.2\approx10699 \) (but wait, wait, no, wait: Wait, the options are A. 6020, B. 8600, C. 8916, D. 10699. Wait, maybe I miscalculated \( (1.2)^{12} \)? Wait, no, let's recalculate \( (1.2)^{12} \):

\( 1.2^1 = 1.2 \)

\( 1.2^2 = 1.44 \)

\( 1.2^3 = 1.728 \)

\( 1.2^4 = 2.0736 \)

\( 1.2^5 = 2.48832 \)

\( 1.2^6 = 2.985984 \)

\( 1.2^7 = 3.5831808 \)

\( 1.2^8 = 4.29981696 \)

\( 1.2^9 = 5.159780352 \)

\( 1.2^{10} = 6.1917364224 \)

\( 1.2^{11} = 7.43008370688 \)

\( 1.2^{12} = 8.916100448256 \)

Then \( y = 1200\times8.916100448256\approx1200\times8.916 = 10699.2 \), which is approximately 10699. Wait, but option D is 10,699. But wait, maybe I made a mistake in the regression equation? Wait, the regression equation is \( y = 1200(1.2)^x \). Let's check with x=1: \( 1200\times1.2 = 1440 \), which matches the table (x=1, y=1440). x=2: \( 1200\times1.44 = 1728 \), which matches the table (x=2, y=1728). So the equation is correct. Then for x=12, \( y = 1200\times(1.2)^{12}\approx1200\times8.916 = 10699 \), which is option D. Wait, but the option C is 8916, which is \( 1200\times7.43 = 8916 \) (x=11), but x=12. So the correct calculation gives approximately 10699, which is option D.

Wait, but let's re-express the steps:

1. Given \( y = 1200(1.2)^x \), x=12.

1. Compute \( (1.2)^{12} \approx 8.916 \) (as above).

1. Multiply by 1200: \( 1200 \times 8.916 = 10699.2 \approx 10699 \).

So the best prediction is 10,699, which is option D.

Answer:

D. 10,699