To determine the predicted profit in the year 4 using the given quadratic regression equation [tex]\( y = -0.34x^2 + 4.43x + 3.46 \)[/tex], we need to follow these steps:
1. Identify the value of [tex]\( x \)[/tex]:
- The year given is 4, so [tex]\( x = 4 \)[/tex].
2. Plug the value of [tex]\( x \)[/tex] into the quadratic regression equation:
- The quadratic regression equation given is [tex]\( y = -0.34x^2 + 4.43x + 3.46 \)[/tex].
3. Substitute [tex]\( x = 4 \)[/tex] into the equation:
- Replace [tex]\( x \)[/tex] with 4 in the equation:
[tex]\[
y = -0.34(4)^2 + 4.43(4) + 3.46
\][/tex]
4. Calculate each term separately:
- First term [tex]\( -0.34(4)^2 \)[/tex]:
[tex]\[
4^2 = 16
\][/tex]
[tex]\[
-0.34 \times 16 = -5.44
\][/tex]
- Second term [tex]\( 4.43(4) \)[/tex]:
[tex]\[
4.43 \times 4 = 17.72
\][/tex]
- Combine all terms together:
[tex]\[
y = -5.44 + 17.72 + 3.46
\][/tex]
5. Add the results:
- Adding the terms together:
[tex]\[
-5.44 + 17.72 = 12.28
\][/tex]
[tex]\[
12.28 + 3.46 = 15.74
\][/tex]
6. Determine the predicted profit:
- The predicted profit for the year 4 is [tex]\( \$ 15.74 \)[/tex] million.
Among the given options, the correct prediction of the profit in year 4 is:
B. [tex]$\$[/tex] 15.74$ million
