To determine the number of new cases in year 15 using the quadratic regression equation [tex]\( y = -2x^2 + 36x + 6 \)[/tex]:
1. Substitute [tex]\( x = 15 \)[/tex] into the equation:
[tex]\[
y = -2(15)^2 + 36(15) + 6
\][/tex]
2. Calculate [tex]\( 15^2 \)[/tex]:
[tex]\[
15^2 = 225
\][/tex]
3. Multiply [tex]\( 225 \)[/tex] by [tex]\(-2\)[/tex]:
[tex]\[
-2 \times 225 = -450
\][/tex]
4. Multiply [tex]\( 15 \)[/tex] by [tex]\( 36 \)[/tex]:
[tex]\[
36 \times 15 = 540
\][/tex]
5. Add these values to the constant term [tex]\( 6 \)[/tex]:
[tex]\[
y = -450 + 540 + 6
\][/tex]
6. Perform the additions:
[tex]\[
y = 90 + 6
\][/tex]
[tex]\[
y = 96
\][/tex]
Hence, the best prediction for the number of new cases in year 15 is [tex]\( \boxed{96} \)[/tex].
