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 20?
A. 618
B. 422
C. 234
D. 88
Answer (2)
Substitute x = 20 into the equation y = − 2 x 2 + 44 x + 8 .
Calculate y = − 2 ( 20 ) 2 + 44 ( 20 ) + 8 .
Simplify the expression: y = − 800 + 880 + 8 .
The predicted number of new cases in year 20 is 88 .
Explanation
Understanding the Problem We are given the quadratic regression equation y = − 2 x 2 + 44 x + 8 , which models the number of cases of a new disease, where x represents the year. We want to predict the number of new cases in year 20.
Substituting the Value of x To find the predicted 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 can be used to model various real-world phenomena, such as the trajectory of a ball, the growth of a population, or the sales of a product over time. In this case, we used a quadratic regression equation to model the number of cases of a new disease. By predicting the number of cases in a future year, public health officials can better prepare for and respond to the outbreak.
The predicted number of new cases of the disease in year 20 is 88, calculated by substituting 20 into the quadratic regression equation. Upon simplification, the outcome confirms that option D, 88, is correct.
;
