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The number of cases of a new disease can be modeled by the quadratic regression equation [tex]\( y = -2x^2 + 36x + 6 \)[/tex], where [tex]\( x \)[/tex] represents the year.

Which is the best prediction for the number of new cases in year 15?

A. 328
B. 194
C. 614
D. 96



Answer :

GinnyAnswer
To predict the number of new cases of the disease in year 15 using the quadratic regression equation [tex]\( y = -2x^2 + 36x + 6 \)[/tex], we need to follow these steps:

1. Identify the given equation and the year for which the prediction is required:
The equation is:
[tex]\[ y = -2x^2 + 36x + 6 \][/tex]
We need to predict the number of new cases when [tex]\( x = 15 \)[/tex].

2. Substitute [tex]\( x = 15 \)[/tex] into the equation:
[tex]\[ y = -2(15)^2 + 36(15) + 6 \][/tex]

3. Calculate each term step-by-step:
- First, compute [tex]\( 15^2 \)[/tex]:
[tex]\[ 15^2 = 225 \][/tex]
- Next, multiply this result by [tex]\(-2\)[/tex]:
[tex]\[ -2 \times 225 = -450 \][/tex]
- Now, compute [tex]\( 36 \times 15 \)[/tex]:
[tex]\[ 36 \times 15 = 540 \][/tex]
- Finally, add the constant term [tex]\( 6 \)[/tex] to the sum of the previous results:
[tex]\[ y = -450 + 540 + 6 \][/tex]

4. Combine the results:
[tex]\[ y = 90 + 6 \][/tex]

5. Calculate the final [tex]\( y \)[/tex] value:
[tex]\[ y = 96 \][/tex]

Therefore, the best prediction for the number of new cases in year 15 is:
[tex]\[ \boxed{96} \][/tex]

So, the correct answer is D: 96.

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