In the year 2000, the population of Ohio was 11.03 million people. By the year 2017, the population of Ohio was 11.66 million people. Create an equation modeling the population, p, of Ohio given the year, t. Round to the nearest hundredth if necessary.
A. [tex]$p =11.1 t +232.14$[/tex]
B. [tex]$p =11.1 t -11.66$[/tex]
C. [tex]$p =0.09 t +191.93$[/tex]
D. [tex]$p =0.04 x -68.97$[/tex]
Answer (2)
Calculate the slope using the two given points: m = 2017 − 2000 11.66 − 11.03 = 0.0370588235... ≈ 0.04 .
Calculate the y-intercept using the point-slope form and the rounded slope: b = 11.03 − 0.04 ∗ 2000 = − 68.97 .
Form the equation: p = 0.04 x − 68.97 .
The equation modeling the population of Ohio is: p = 0.04 x − 68.97
Explanation
Understanding the Problem We are given the population of Ohio in the years 2000 and 2017 and asked to create a linear equation modeling the population, p , as a function of the year, L .
Identifying Data Points We have two data points: ( 2000 , 11.03 ) and ( 2017 , 11.66 ) . We will use these points to determine the slope and y-intercept of the linear equation.
Calculating the Slope First, we calculate the slope, m , using the formula: m = L 2 − L 1 p 2 − p 1 = 2017 − 2000 11.66 − 11.03 = 17 0.63 = 0.0370588235... Rounding to the nearest hundredth gives us m ≈ 0.04 .
Calculating the Y-Intercept Next, we calculate the y-intercept, b , using the point-slope form of a line and one of the given points. We'll use the point ( 2000 , 11.03 ) :
p − p 1 = m ( L − L 1 ) 11.03 = 0.0370588235 ∗ 2000 + b b = 11.03 − 0.0370588235 ∗ 2000 = 11.03 − 74.117647 = − 63.087647 Rounding to the nearest hundredth gives us b ≈ − 63.09 .
Forming the Equation Now we can write the equation in slope-intercept form: p = m L + b p = 0.04 L − 63.09 However, the options given use t or x instead of L as the variable for the year. Also, the options provide a different constant term. Let's re-evaluate the y-intercept using the rounded slope m = 0.04 :
b = 11.03 − 0.04 ∗ 2000 = 11.03 − 80 = − 68.97 So the equation is: p = 0.04 L − 68.97 Replacing L with x , we get: p = 0.04 x − 68.97
Final Answer Comparing our equation with the given options, we see that it matches option d. Therefore, the equation modeling the population of Ohio is: p = 0.04 x − 68.97
Examples
Linear equations can be used to model population growth, predict future trends, and make informed decisions about resource allocation and planning. For example, city planners can use population models to estimate the need for new schools, hospitals, and infrastructure. Businesses can use these models to forecast demand for their products and services. Understanding linear models helps in making data-driven decisions in various real-world scenarios.
To model Ohio's population from 2000 to 2017, we calculated a slope of approximately 0.04 and a y-intercept of -68.97. The resulting equation is p = 0.04 x − 68.97 . Thus, the correct choice from the options provided is D.
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