Math Problem Statement

The chart to the right shows a​ country's annual egg production. Model the data in the chart with a linear​ function, using the points ​(1994​,52.7​) and ​(1998​,61.4​). Let x represent the​ year, where xequals0 represents 1994​, xequals1 represents 1995​, and so​ on, and let y represent the egg production​ (in billions). Predict egg production in 2000. Year Egg production ​(in billions) 1994 52.7 1995 53.5 1996 55.4 1997 58.3 1998 61.4 1999 65 2000 70.9 Question content area bottom Part 1 The linear model for the data is    enter your response here. ​(Type your answer in​ slope-intercept form. Use integers or decimals for any numbers in the equation. Round to three decimal places as​ needed.)

Solution

To find the linear model in slope-intercept form y=mx+by = mx + b, we first determine the slope mm using the two given points:
(1994,52.7)and(1998,61.4)(1994, 52.7) \quad \text{and} \quad (1998, 61.4) Since x=0x = 0 corresponds to 1994, we redefine the points as: (0,52.7)and(4,61.4)(0, 52.7) \quad \text{and} \quad (4, 61.4)

Step 1: Calculate the Slope

The slope formula is: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}

m=61.452.740=8.74=2.175m = \frac{61.4 - 52.7}{4 - 0} = \frac{8.7}{4} = 2.175

Step 2: Find the y-intercept

Using the slope-intercept form y=mx+by = mx + b and the point (0,52.7)(0, 52.7):

52.7=2.175(0)+b52.7 = 2.175(0) + b

b=52.7b = 52.7

Step 3: Write the Equation

y=2.175x+52.7y = 2.175x + 52.7

Step 4: Predict Egg Production in 2000

For the year 2000, we set x=6x = 6:

y=2.175(6)+52.7y = 2.175(6) + 52.7

y=13.05+52.7=65.75y = 13.05 + 52.7 = 65.75

So, the predicted egg production in 2000 is 65.75 billion eggs (rounded to three decimal places: 65.750).

Final Answer:

Linear model:
y=2.175x+52.7y = 2.175x + 52.7

Predicted production in 2000:
65.750 billion eggs65.750 \text{ billion eggs}

Would you like a further breakdown or another method?


Related Questions:

  1. How do you interpret the slope of this linear function in real-world terms?
  2. If the trend continues, what would be the predicted egg production in 2005?
  3. How well does this linear model fit the actual data for 1999 and 2000?
  4. What are some potential limitations of using a linear model for this prediction?
  5. How would the equation change if we set x=0x = 0 to represent 1990 instead?

Tip:

A linear model assumes a constant rate of increase. If the actual data shows acceleration, a quadratic or exponential model may be more accurate.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Slope-Intercept Form
Linear Modeling

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Slope-intercept form: y = mx + b

Theorems

Slope formula for linear functions

Suitable Grade Level

Grades 7-9