If the culture triples every 5 hours, what should $B(5)$ be in terms of the starting amount? How does that relate to the exponent in an exponential function?

In an exponential model, the base (here related to 3) tells you the growth factor, and the exponent controls how fast it’s applied. Which exponent expression in t makes the population multiply by 3 exactly when t increases by 5?

Try plugging t = 5 into each choice. Which one gives a value exactly three times larger than the value at t = 0?

Type each option into Desmos as a separate function, for example:

• B1(t) = 500*(3*t)^5
• B2(t) = 500*3^(t/5)
• B3(t) = 500*3^(5*t)
• B4(t) = 500*5^(3*t) Make sure you use parentheses so the exponents and products are interpreted correctly.

For each function, either tap on the graph at t = 0 or use a table (tap the gear icon and select "Convert to table"). Look at the y-value when t = 0. Only the function whose y-value is 500 at t = 0 can model the situation correctly.

Still using the graphs or tables, look at the y-value when t = 5 for the functions that had 500 at t = 0. The correct model will give a value exactly three times larger than 500 at t = 5 (because the culture triples every 5 hours).

Optionally, check t = 10 and t = 15 for that same function. You should see the value triple each time t increases by 5 hours, confirming it matches the described growth pattern.

The bacteria population triples in equal time intervals (every 5 hours), which is a constant multiplicative change. That means this is an exponential growth situation, not linear or polynomial growth. So the model should look like an initial amount times a base raised to a power involving t.

At time t = 0, the culture starts with 500 cells. Any correct model must satisfy $B(0) = 500$.

• For an exponential model, this looks like $B(t) = 500 \times (\text{something involving } t)$, or more specifically $B(t) = 500 \cdot b^{f(t)}$ for some base b and exponent f(t).
• So the 500 should be the factor in front, not mixed with t inside a power or parentheses in a way that makes $B(0)$ something other than 500.

“Triples every 5 hours” means:

• At t = 0: $B(0) = 500$.
• At t = 5: $B(5) = 3 \times 500 = 1500$.
• At t = 10: $B(10) = 3 \times 1500 = 4500$, and so on.

In an exponential model of the form $B(t) = 500 \cdot 3^{f(t)}$, we want the exponent to increase by 1 every 5 hours, because each step of 1 in the exponent multiplies the amount by 3. That means the exponent should be $t/5$:

• When t = 0, exponent = 0.
• When t = 5, exponent = 1.
• When t = 10, exponent = 2, etc.

Using base 3 and exponent $t/5$, the model is $B(t) = 500(3)^{t/5}$.

• This gives $B(0) = 500(3)^{0} = 500$.
• It also gives $B(5) = 500(3)^{1} = 1500$, which is triple the initial amount, and continues to triple every 5 hours.

So the correct choice is $B(t)=500(3)^{t/5}$, which is answer choice B.