The number of cases of a new disease can be modeled by the quadratic regression equation [tex]y=-2 x^2+44 x+8[/tex], where [tex]x[/tex] represents the year.
Which is the best prediction for the number of new cases in year [tex]20[/tex]?
A. 618
B. 422
C. 234
D. 88
Answer (1)
Substitute x = 20 into the quadratic regression equation: y = − 2 x 2 + 44 x + 8 .
Calculate y = − 2 ( 20 ) 2 + 44 ( 20 ) + 8 .
Simplify the equation: y = − 800 + 880 + 8 .
Determine the predicted number of new cases: 88 .
Explanation
Understanding the Problem We are given a quadratic equation that models the number of cases of a new disease, where x represents the year and y represents the number of cases. The equation is: y = − 2 x 2 + 44 x + 8 We want to predict the number of new cases in year 20.
Substituting the Value of x To predict the number of new cases in year 20, we substitute x = 20 into the equation: y = − 2 ( 20 ) 2 + 44 ( 20 ) + 8
Calculating the Value of y Now, we calculate the value of y :
y = − 2 ( 400 ) + 880 + 8 y = − 800 + 880 + 8 y = 80 + 8 y = 88
Final Answer Therefore, the best prediction for the number of new cases in year 20 is 88.
Examples
Quadratic regression equations are used in epidemiology to model the spread of diseases. For example, public health officials can use these models to predict the number of cases in future years, which helps them allocate resources and implement interventions. Understanding how to substitute values into a quadratic equation allows for informed decision-making in public health planning.
