Question

The number of bacteria in a culture is increasing according to the law of exponential growth. After 2 hours, there are 50 bacteria, and after 5 hours, there are 400 bacteria. How many bacteria will there be after 7 hours?

Answer 1

Given:

The number of bacteria in a culture is increasing according to the law of exponential growth.

After 2 hours, there are 50 bacteria, and after 5 hours, there are 400 bacteria.

To find:

The number of bacteria after 7 hours.

Solution:

Let y be the number of bacteria after x hours.

The general exponential growth model is

$y=ab^x$

where, a is initial value and b is growth factor.

After 2 hours, there are 50 bacteria. So, the point is (2,50). Put x=2 and y=50.

$50=ab^2$         ...(ii)

After 5 hours, there are 400 bacteria. So, the point is (5,400). Put x=5 and y=400 in general exponential growth model.

$400=ab^5$       ...(iii)

Divide equation (iii) by (ii).

$\dfrac{400}{50}=\dfrac{ab^5}{ab^2}$  

$8=b^3$

Taking cube root on both sides, we get

$b=2$

Put b=2 in (ii).

$50=a(2)^2$  

$50=4a$

$\dfrac{50}{4}=a$

$a=12.5$

Put a=12.5 and b=2 in (i).

$y=12.5(2)^x$

This is the equation which represents number of bacteria after x hours.

Put x=7 in the above equation.

$y=12.5(2)^7$

$y=12.5(128)$

$y=1600$

Therefore, the number of bacteria after 7 hours is 1600.