The problem presents the exponential function $P(t) = 100(2)^t$, which models the population of a bacterial colony $t$ hours after the start of an experiment. We need to interpret the point $(3, 800)$.

Analysis

1. Given Function: $P(t) = 100(2)^t$

   • $P(t)$: Population of the bacteria at time $t$.
   • $t$: Time in hours.

2. Interpretation of the Point $(3, 800)$:

   • The point $(3, 800)$ indicates that at $t = 3$ (3 hours after the start), the population $P(3)$ equals 800.

3. Verification:

   • Substituting $t = 3$: $P(3) = 100(2)^3 = 100 \times 8 = 800$
   • This confirms that after 3 hours, the population reaches 800.

Correct Answer

The correct interpretation is: After 3 hours, the population of the bacterial colony has increased to 800. This matches the selected option.

Would you like more details or have any questions about this problem?

Related Questions:

1. How would the interpretation change if the exponential base were different, such as $100(3)^t$?
2. What does the base $2$ represent in this exponential model?
3. How do you determine the growth rate from an exponential function like $P(t) = 100(2)^t$?
4. What would the population be after 5 hours using the same function?
5. How can you determine when the population reaches 1600?

Tip:

In exponential growth models, the base of the exponent indicates how the quantity changes for each unit increase in $t$. A base of 2 means the quantity doubles each time $t$ increases by 1.