Question

A scientist running an experiment starts with 100 bacteria cells. These bacteria double their population every 15 hours. Find how long it takes for
the bacteria cells to increase to 300. Use the formula Pt = P02t/d, where P0 is the original number of bacteria cells, Pt is the number
after t hours, and d is the time taken to double the number.
It takes ( )
hours for the number of bacteria to increase to 300.

Answer 1

Answer:

About 24 hours.

Step-by-step explanation:

We are given that the experiment initially had 100 bacteria cells, and that its populations double every 15 hours.

From the given formula:

$\displaystyle P_t=P_0(2)^\dfrac{t}{d}$

We can substitute 100 for P₀ and 15 for t. This yields:

$\displasytyle P_t=100(2)^\dfrac{t}{15}$

We want to find after how many hours t will the population reach 300.

So, let's substitute 300 for P_t and solve for t:

$\displaystyle 300=100(2)^\dfrac{t}{15}$

Divide:

$3=(2)^\dfrac{t}{15}$

We can take the natural log of both sides:

$\ln(3)=\ln\left(2^\dfrac{t}{15}\right)$

Logarithm properties:

$\displaystyle \ln(3)=\frac{t}{15}\ln(2)$

Solve for t:

$\displaystyle t=\frac{15\ln(3)}{\ln(2)}$

Use a calculator:

$t\approx23.77$

In conclusion, the population will reach 300 bacteria after about 24 hours.