Growth of a bacteria population $$ B(t) $$ (in millions), at time $$ t $$ (hours), follows the logisic growth $$ B(t)=\frac{4}{1+7 e^{-t}}, \quad t \geq 0 $$ (a) Sketch the graph of the bacteria population verses time, for the first eight hours. (b) What is the initial population size? (d) What value does the bacteria population approach after large time?

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To analyze the logistic growth of the bacteria population given by the function

$$
B(t) = \frac{4}{1 + 7 e^{-t}}, \quad t \geq 0,
$$

let's address each part of the question.

### (a) Sketch the graph of the bacteria population versus time for the first eight hours.

To sketch the graph, we can calculate the values of $$ B(t) $$ at several key points from $$ t = 0 $$ to $$ t = 8 $$.

1. **At $$ t = 0 $$**:
   $$
   B(0) = \frac{4}{1 + 7 e^{0}} = \frac{4}{1 + 7} = \frac{4}{8} = 0.5 \text{ million}
   $$

2. **At $$ t = 1 $$**:
   $$
   B(1) = \frac{4}{1 + 7 e^{-1}} \approx \frac{4}{1 + 7 \cdot 0.3679} \approx \frac{4}{1 + 2.5753} \approx \frac{4}{3.5753} \approx 1.12 \text{ million}
   $$

3. **At $$ t = 2 $$**:
   $$
   B(2) = \frac{4}{1 + 7 e^{-2}} \approx \frac{4}{1 + 7 \cdot 0.1353} \approx \frac{4}{1 + 0.9471} \approx \frac{4}{1.9471} \approx 2.05 \text{ million}
   $$

4. **At $$ t = 3 $$**:
   $$
   B(3) = \frac{4}{1 + 7 e^{-3}} \approx \frac{4}{1 + 7 \cdot 0.0498} \approx \frac{4}{1 + 0.3486} \approx \frac{4}{1.3486} \approx 2.96 \text{ million}
   $$

5. **At $$ t = 4 $$**:
   $$
   B(4) = \frac{4}{1 + 7 e^{-4}} \approx \frac{4}{1 + 7 \cdot 0.0183} \approx \frac{4}{1 + 0.1281} \approx \frac{4}{1.1281} \approx 3.55 \text{ million}
   $$

6. **At $$ t = 5 $$**:
   $$
   B(5) = \frac{4}{1 + 7 e^{-5}} \approx \frac{4}{1 + 7 \cdot 0.0067} \approx \frac{4}{1 + 0.0469} \approx \frac{4}{1.0469} \approx 3.82 \text{ million}
   $$

7. **At $$ t = 6 $$**:
   $$
   B(6) = \frac{4}{1 + 7 e^{-6}} \approx \frac{4}{1 + 7 \cdot 0.0025} \approx \frac{4}{1 + 0.0175} \approx \frac{4}{1.0175} \approx 3.93 \text{ million}
   $$

8. **At $$ t = 7 $$**:
   $$
   B(7) = \frac{4}{1 + 7 e^{-7}} \approx \frac{4}{1 + 7 \cdot 0.000911} \approx \frac{4}{1 + 0.006377} \approx \frac{4}{1.006377} \approx 3.97 \text{ million}
   $$

9. **At $$ t = 8 $$**:
   $$
   B(8) = \frac{4}{1 + 7 e^{-8}} \approx \frac{4}{1 + 7 \cdot 0.000335} \approx \frac{4}{1 + 0.002345} \approx \frac{4}{1.002345} \approx 3.99 \text{ million}
   $$

Now, we can plot these points to sketch the graph. The graph will start at $$ 0.5 $$ million at $$ t = 0 $$ and will approach $$ 4 $$ million as $$ t $$ increases.

### (b) What is the initial population size?

The initial population size is given by $$ B(0) $$:
$$
B(0) = 0.5 \text{ million}.
$$

### (c) What value does the bacteria population approach after a large time?

As $$ t $$ approaches infinity, the term $$ e^{-t} $$ approaches $$ 0 $$. Therefore, we can find the limit of $$ B(t) $$ as follows:

$$
\lim_{t \to \infty} B(t) = \frac{4}{1 + 7 \cdot 0} = \frac{4}{1} = 4 \text{ million}.
$$

Thus, the bacteria population approaches $$ 4 $$ million as time goes to infinity.

### Summary:
- (a) The graph shows a sigmoidal shape starting at $$ 0.5 $$ million and approaching $$ 4 $$ million.
- (b) The initial population size is $$ 0.5 $$ million.
- (c) The population approaches $$ 4 $$ million as time increases.
(a) Sketch the graph of the bacteria population versus time for the first eight hours. The graph starts at 0.5 million and approaches 4 million. (b) The initial population size is 0.5 million. (c) The population approaches 4 million as time goes to infinity.

Growth of a bacteria population (in millions), at time (hours), follows the logisic
growth

(a) Sketch the graph of the bacteria population verses time, for the first eight hours.
(b) What is the initial population size?
(d) What value does the bacteria population approach after large time?

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Growth of a bacteria population (in millions), at time (hours), follows the logisic growth (a) Sketch the graph of the bacteria population verses time, for the first eight hours. (b) What is the initial population size? (d) What value does the bacteria population approach after large time?