Alright, let's go through the problem step-by-step to find the correct expression for the number of bacteria in terms of the number of hours [tex]\( h \)[/tex] the food has been unrefrigerated.
We start with the following:
1. The number of bacteria [tex]\( b \)[/tex] is given by:
[tex]\[ b(t) = 20t^2 - 70t + 300 \][/tex]
where [tex]\( t \)[/tex] is the temperature in degrees Fahrenheit.
2. The temperature in terms of hours [tex]\( h \)[/tex] is given by:
[tex]\[ t(h) = 2h + 3 \][/tex]
To find the number of bacteria [tex]\( b \)[/tex] as a function of [tex]\( h \)[/tex], we need to substitute the expression for [tex]\( t \)[/tex] into the equation for [tex]\( b(t) \)[/tex].
Let's do this substitution step-by-step:
- Substitute [tex]\( t = 2h + 3 \)[/tex] into [tex]\( b(t) \)[/tex]:
[tex]\[ b(t(h)) = 20(2h + 3)^2 - 70(2h + 3) + 300 \][/tex]
Next, we need to expand the expression [tex]\( (2h + 3)^2 \)[/tex]:
- Expand [tex]\( (2h + 3)^2 \)[/tex] using the algebraic identity [tex]\( (a+b)^2 = a^2 + 2ab + b^2 \)[/tex]:
[tex]\[ (2h + 3)^2 = (2h)^2 + 2 \cdot 2h \cdot 3 + 3^2 \][/tex]
[tex]\[ (2h + 3)^2 = 4h^2 + 12h + 9 \][/tex]
Now, substitute [tex]\( 4h^2 + 12h + 9 \)[/tex] back into the bacteria function:
[tex]\[ b(2h + 3) = 20(4h^2 + 12h + 9) - 70(2h + 3) + 300 \][/tex]
Expand and simplify each term:
1. Distribute 20:
[tex]\[ 20(4h^2 + 12h + 9) = 80h^2 + 240h + 180 \][/tex]
2. Distribute -70:
[tex]\[ -70(2h + 3) = -140h - 210 \][/tex]
3. Combine all terms together, including the constant 300:
[tex]\[ b(2h + 3) = 80h^2 + 240h + 180 - 140h - 210 + 300 \][/tex]
Now, combine like terms:
1. Combine the [tex]\( h^2 \)[/tex] terms:
[tex]\[ 80h^2 \][/tex]
2. Combine the [tex]\( h \)[/tex] terms:
[tex]\[ 240h - 140h = 100h \][/tex]
3. Combine the constant terms:
[tex]\[ 180 - 210 + 300 = 270 \][/tex]
So, the 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]\[ b(h) = 80h^2 + 100h + 270 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{80h^2 + 100h + 270} \][/tex]
which corresponds to option B.
