To determine the function that describes the difference in the number of bacteria, [tex]\( N(t) \)[/tex], for both species after [tex]\( t \)[/tex] hours, we need to subtract the function representing the bacteria of species [tex]\( B \)[/tex] from the function representing the bacteria of species [tex]\( A \)[/tex].
Given:
- The number of bacteria of species [tex]\( A \)[/tex] after [tex]\( t \)[/tex] hours is [tex]\( A(t) = 5 + (0.25t)^3 \)[/tex]
- The number of bacteria of species [tex]\( B \)[/tex] after [tex]\( t \)[/tex] hours is [tex]\( B(t) = 2 + 8(1.06)^t \)[/tex]
The difference in the number of bacteria between the two species is:
[tex]\[ N(t) = A(t) - B(t) \][/tex]
Substituting the given functions into the equation:
[tex]\[ N(t) = \left( 5 + (0.25t)^3 \right) - \left( 2 + 8(1.06)^t \right) \][/tex]
Now, distribute the subtraction:
[tex]\[ N(t) = 5 + (0.25t)^3 - 2 - 8(1.06)^t \][/tex]
Simplify the constants:
[tex]\[ N(t) = (0.25t)^3 + 5 - 2 - 8(1.06)^t \][/tex]
[tex]\[ N(t) = (0.25t)^3 + 3 - 8(1.06)^t \][/tex]
Reordering the expression gives us:
[tex]\[ N(t) = 3 + (0.25t)^3 - 8(1.06)^t \][/tex]
Comparing this result with the given options, we see that the correct option is:
[tex]\[ \boxed{D \quad N(t) = 3 + (0.25t)^3 - 8(1.06)^t} \][/tex]
