Math Snap

STEP 1

1. The function $f(t) = 60,000(2)^{\frac{t}{40}}$ represents the number of bacteria at time $t$ minutes.
2. We need to find the time $t$ when the number of bacteria doubles from its initial amount.

STEP 2

1. Determine the initial number of bacteria.
2. Set up an equation for the doubled bacteria count.
3. Solve the equation for $t$.

STEP 3

Determine the initial number of bacteria.
The initial number of bacteria is given by $f(0)$.
$$f(0) = 60,000(2)^{\frac{0}{40}} = 60,000 \times 1 = 60,000$$

STEP 4

Set up an equation for the doubled bacteria count.
The bacteria count doubles when it reaches $2 \times 60,000 = 120,000$.
Set $f(t) = 120,000$.
$$60,000(2)^{\frac{t}{40}} = 120,000$$

Solve the equation for $t$.
Divide both sides by 60,000 to isolate the exponential term:
$$(2)^{\frac{t}{40}} = \frac{120,000}{60,000}$$ $$(2)^{\frac{t}{40}} = 2$$ Since $2^1 = 2$, we have:
$$\frac{t}{40} = 1$$ Multiply both sides by 40 to solve for $t$:
$$t = 40 \times 1$$ $$t = 40$$ It takes $\boxed{40}$ minutes for the number of bacteria to double.