Answer :
To solve this problem, we need to create a linear model using the provided data points and then use that model to predict egg production for the year 2000. Let's go through the steps:
1. Identify the Points:
We have two points given in the problem:
- Point 1: [tex]\((0, 51.7)\)[/tex], corresponding to the year 1994.
- Point 2: [tex]\((4, 60.3)\)[/tex], corresponding to the year 1998.
2. Calculate the Slope (m):
The slope [tex]\(m\)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated with the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points:
[tex]\[
m = \frac{60.3 - 51.7}{4 - 0} = \frac{8.6}{4} = 2.150
\][/tex]
3. Find the y-intercept (b):
To find the y-intercept [tex]\(b\)[/tex], we use the equation of a line in slope-intercept form, [tex]\(y = mx + b\)[/tex]. We can substitute the slope and one of the points into this equation to solve for [tex]\(b\)[/tex]. Using the point [tex]\((0, 51.7)\)[/tex]:
[tex]\[
51.7 = 2.150 \cdot 0 + b
\][/tex]
Solving for [tex]\(b\)[/tex],
[tex]\[
b = 51.7
\][/tex]
4. Write the Equation of the Line:
The equation of the linear model is:
[tex]\[
y = 2.150x + 51.700
\][/tex]
5. Predict Egg Production in 2000:
The year 2000 corresponds to [tex]\(x = 6\)[/tex]. We substitute [tex]\(x = 6\)[/tex] into the linear equation to find the predicted egg production:
[tex]\[
y = 2.150 \cdot 6 + 51.700 = 12.9 + 51.7 = 64.6
\][/tex]
6. Compare with Actual Data:
The predicted egg production for the year 2000 is 64.6 billion eggs. The actual data for the year 2000 shows an egg production of 69.7 billion eggs. The prediction is slightly lower than the actual production by about 5.1 billion eggs.
So, the linear model is [tex]\(y = 2.150x + 51.700\)[/tex] and the predicted egg production for 2000 is 64.6 billion eggs.
1. Identify the Points:
We have two points given in the problem:
- Point 1: [tex]\((0, 51.7)\)[/tex], corresponding to the year 1994.
- Point 2: [tex]\((4, 60.3)\)[/tex], corresponding to the year 1998.
2. Calculate the Slope (m):
The slope [tex]\(m\)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated with the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points:
[tex]\[
m = \frac{60.3 - 51.7}{4 - 0} = \frac{8.6}{4} = 2.150
\][/tex]
3. Find the y-intercept (b):
To find the y-intercept [tex]\(b\)[/tex], we use the equation of a line in slope-intercept form, [tex]\(y = mx + b\)[/tex]. We can substitute the slope and one of the points into this equation to solve for [tex]\(b\)[/tex]. Using the point [tex]\((0, 51.7)\)[/tex]:
[tex]\[
51.7 = 2.150 \cdot 0 + b
\][/tex]
Solving for [tex]\(b\)[/tex],
[tex]\[
b = 51.7
\][/tex]
4. Write the Equation of the Line:
The equation of the linear model is:
[tex]\[
y = 2.150x + 51.700
\][/tex]
5. Predict Egg Production in 2000:
The year 2000 corresponds to [tex]\(x = 6\)[/tex]. We substitute [tex]\(x = 6\)[/tex] into the linear equation to find the predicted egg production:
[tex]\[
y = 2.150 \cdot 6 + 51.700 = 12.9 + 51.7 = 64.6
\][/tex]
6. Compare with Actual Data:
The predicted egg production for the year 2000 is 64.6 billion eggs. The actual data for the year 2000 shows an egg production of 69.7 billion eggs. The prediction is slightly lower than the actual production by about 5.1 billion eggs.
So, the linear model is [tex]\(y = 2.150x + 51.700\)[/tex] and the predicted egg production for 2000 is 64.6 billion eggs.
