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.
