To find the correct expression for the number of bacteria [tex]\( b \)[/tex] as a function of the number of hours [tex]\( h \)[/tex] after the food is removed from refrigeration, we need to follow these steps:
1. Understanding the Given Functions:
- The number of bacteria as a function of temperature [tex]\( t \)[/tex] is given by:
[tex]\[
b(t) = 20t^2 - 70t + 300
\][/tex]
- The temperature [tex]\( t \)[/tex] as a function of the number of hours [tex]\( h \)[/tex] is:
[tex]\[
t(h) = 2h + 3
\][/tex]
2. Substitute the Temperature Function into the Bacteria Function:
- To express the number of bacteria directly as a function of hours [tex]\( h \)[/tex], we substitute [tex]\( t(h) \)[/tex] into [tex]\( b(t) \)[/tex]. This means we need to replace [tex]\( t \)[/tex] in the bacteria function with [tex]\( 2h + 3 \)[/tex].
[tex]\[
b(h) = 20(2h + 3)^2 - 70(2h + 3) + 300
\][/tex]
3. Expand and Simplify the Expression:
- First, expand [tex]\( (2h + 3)^2 \)[/tex]:
[tex]\[
(2h + 3)^2 = 4h^2 + 12h + 9
\][/tex]
- Now substitute this expansion into the bacteria function:
[tex]\[
b(h) = 20(4h^2 + 12h + 9) - 70(2h + 3) + 300
\][/tex]
[tex]\[
= 20 \cdot 4h^2 + 20 \cdot 12h + 20 \cdot 9 - 70 \cdot 2h - 70 \cdot 3 + 300
\][/tex]
[tex]\[
= 80h^2 + 240h + 180 - 140h - 210 + 300
\][/tex]
4. Combine Like Terms:
- Combine the terms involving [tex]\( h \)[/tex]:
[tex]\[
80h^2 + (240h - 140h) + (180 - 210 + 300)
\][/tex]
[tex]\[
= 80h^2 + 100h + 270
\][/tex]
Therefore, the correct expression representing the number of bacteria [tex]\( b \)[/tex] in the food as a function of the number of hours [tex]\( h \)[/tex] the food is unrefrigerated is:
[tex]\[
\boxed{80h^2 + 100h + 270}
\][/tex]
Thus, the correct answer is:
[tex]\[
\text{B. } 80h^2 + 100h + 270
\][/tex]
