Let's break down the calculation step by step and find the correct labeling for the given equation, finding the number of bacteria in the food in [tex]$h$[/tex] hours.
We have two functions:
1. The number of bacteria in terms of temperature [tex]\( t \)[/tex]:
[tex]\[
b(t) = 15t^2 - 60t + 125
\][/tex]
2. The temperature in terms of hours [tex]\( h \)[/tex]:
[tex]\[
t(h) = 3h + 4
\][/tex]
Firstly, we are going to substitute [tex]\( t(h) \)[/tex] into [tex]\( b(t) \)[/tex] to find [tex]\( b(t(h)) \)[/tex].
Given:
[tex]\[
t(h) = 3h + 4
\][/tex]
We substitute [tex]\( t(h) \)[/tex] into [tex]\( b(t) \)[/tex]:
[tex]\[
b(t(h)) = b(3h + 4)
\][/tex]
Now, we substitute [tex]\( 3h + 4 \)[/tex] into the equation for bacteria:
[tex]\[
b(3h + 4) = 15(3h + 4)^2 - 60(3h + 4) + 125
\][/tex]
Next, expand and simplify the expression:
[tex]\[
(3h + 4)^2 = (3h)^2 + 2 \cdot 3h \cdot 4 + 4^2 = 9h^2 + 24h + 16
\][/tex]
Substitute this back into the bacterial function:
[tex]\[
b(3h + 4) = 15(9h^2 + 24h + 16) - 60(3h + 4) + 125
\][/tex]
Distribute the 15 and simplify:
[tex]\[
= 135h^2 + 360h + 240 - 180h - 240 + 125
\][/tex]
Combine like terms:
[tex]\[
= 135h^2 + 180h + 125
\][/tex]
Therefore, the function for the number of bacteria in [tex]\( h \)[/tex] hours is:
[tex]\[
b(t(h)) = 135h^2 + 180h + 125
\][/tex]
Using the labels provided, we can fill in the equation accordingly:
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
[tex]$b \circ t(h)$[/tex] & [tex]$=$[/tex] & 135 & [tex]$h^2$[/tex] & + & 180 & [tex]$h$[/tex] & + 125 \\
\hline
\end{tabular}
