QuestionA strain of bacteria doubles every 4 hours. If a sample contains 40 bacteria, determine the time needed until there are 1000 bacteria present. A colony of rates doubles every 3 months. If the colony started at 12 rats, what is the population after 4.5

Studdy Solution

STEP 1

1. The bacteria doubles every 4 hours.
2. The initial number of bacteria is 40.
3. We need to find the time required for the bacteria to reach 1000.

STEP 2

1. Set up the exponential growth formula.
2. Solve for the time needed to reach 1000 bacteria.

STEP 3

Set up the exponential growth formula.
The general formula for exponential growth is: $N(t) = N_0 \times 2^{\frac{t}{T}}$
Where: - $N(t)$ is the number of bacteria at time $t$ . - $N_0$ is the initial number of bacteria. - $T$ is the doubling time. - $t$ is the time elapsed.
For this problem: - $N_0 = 40$ - $T = 4$ hours - $N(t) = 1000$

STEP 4

Solve for the time needed to reach 1000 bacteria.
Substitute the known values into the formula: $1000 = 40 \times 2^{\frac{t}{4}}$
Divide both sides by 40: $25 = 2^{\frac{t}{4}}$
Take the logarithm of both sides to solve for $t$ : $\log_{10}(25) = \frac{t}{4} \log_{10}(2)$
Solve for $t$ : $t = 4 \times \frac{\log_{10}(25)}{\log_{10}(2)}$
Calculate the values: $t \approx 4 \times \frac{1.39794}{0.30103}$ $t \approx 4 \times 4.64386$ $t \approx 18.57544$
The time needed is approximately $\boxed{18.58}$ hours.

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