Let's analyze the problem from the image.

Problem

• At time $t = 0$, 500 bacteria are present in a petri dish.
• The bacteria population triples every 15 days.

The task is to:

1. Give an exponential model for the situation.
2. Determine the number of bacteria after 40 days.

Solution

Part (a) - Exponential Model

For an exponential growth model, the general form is:

$y(t) = y_0 \cdot b^{t/T}$

Where:

• $y(t)$ is the population at time $t$.
• $y_0 = 500$ is the initial population.
• $b = 3$, since the population triples.
• $T = 15$ days, the time it takes for the population to triple.

Thus, the exponential model for this situation is:

$y(t) = 500 \cdot 3^{t/15}$

Part (b) - Population after 40 Days

To find the population after 40 days, we substitute $t = 40$ into the exponential model:

$y(40) = 500 \cdot 3^{40/15}$

First, simplify the exponent:

$40/15 = \frac{8}{3} \approx 2.67$

Now calculate:

$y(40) = 500 \cdot 3^{2.67}$

Using the approximation:

$3^{2.67} \approx 13.99$

So:

$y(40) \approx 500 \cdot 13.99 = 6995$

Thus, the number of bacteria after 40 days is approximately 6,995.

Would you like any further details or clarifications on this?

Additional Questions:

1. How would the model change if the bacteria quadrupled every 10 days?
2. How does the population growth rate affect the shape of the exponential curve?
3. Can you express the growth factor $3^{t/15}$ as a continuous exponential growth function?
4. What would happen if the growth slowed and the bacteria doubled instead of tripled?
5. How can the model be adjusted for a situation where the population decreases over time?

Tip:

Always double-check your exponent calculations when working with exponential models to ensure accuracy!