To determine the correct equation for the bacteria growth after 4 hours, we need to understand how the growth is occurring. In this case, the bacteria quadruple every hour starting from an initial population of 100.
The general formula for exponential growth can be defined as follows:
\[ B(t) = B_0 \cdot r^t \]
Where:
- \( B(t) \) is the population at time \( t \),
- \( B_0 \) is the initial population,
- \( r \) is the growth factor (quadrupling means \( r = 4 \)),
- \( t \) is the time in hours.
For this problem:
- Initial population, \( B_0 = 100 \)
- Growth factor, \( r = 4 \)
- Time, \( t = 4 \) hours
Substituting these values into the formula gives:
\[ B(4) = 100 \cdot 4^4 \]
Now we can check which of the provided answer choices matches this form.
- Option 1: \( B = 100(1 + 4)^4 \) - This option doesn't match our derived formula.
- Option 2: \( B = 4(100)(100)(100)(100) \) - This seems incorrect as it's not in the proper exponential form.
- Option 3: \( B = 4(100)^4 \) - This is also incorrect; it doesn't maintain the proper structure for growth.
- Option 4: \( B = 100(4)^4 \) - This matches our derived formula.
Therefore, the correct equation that matches the number of bacteria in the population after 4 hours is:
\[ B = 100(4^4) \]
Thus, the answer is Option 4: \( B = 100(4^4) \).