Question

The number of bacteria in a culture is increasing according to the law of exponential growth. The initial population is 240 bacteria, and the population after 9 hours is double the population after 1 hour. How many bacteria will there be after 4 hours? (Round your answer to the nearest whole number.)

Answer 1

Answer:

339

Step-by-step explanation:

Exponential Function

General form of an exponential function: $y=ab^x$

where:

• a is the initial value (y-intercept)
• b is the base (growth/decay factor) in decimal form
• x is the independent variable
• y is the dependent variable

If b > 1 then it is an increasing function

If 0 < b < 1 then it is a decreasing function

Given information:

• a = 240 (initial population of bacteria)
• x = time (in hours)
• y = population of bacteria

Therefore:  $y=240b^x$

To find an expression for the population after 1 hour, substitute x = 1 into the found equation:

$\implies y=240b^1$

$\implies y=240b$

We are told that the population after 9 hours is double the population after 1 hour.  Therefore, make y equal to twice the found expression for the population after 1 hour, let x = 9, then solve for b:

$\implies 2(240b)=240b^9$

$\implies 480b=240b^9$

$\implies 480=240b^8$

$\implies 2=b^8$

$\implies b=\sqrt[8]{2}$

$\implies b=2^{\frac{1}{8}}$

Therefore, the final exponential equation modelling the given scenario is:

$\implies y=240(2^{\frac{1}{8}})^x$

$\implies y=240(2)^{\frac{1}{8}x}$

To find how many bacteria there will be after 4 hours, substitute x = 4 into the found equation:

$\implies y=240(2)^{\frac{1}{8}(4)}$

$\implies y=240(2)^{\frac{1}{2}}$

$\implies y=339 \:\: \sf (nearest\:whole\:number)$

Therefore, there will be 339 bacteria (rounded to the nearest whole number) after 4 hours.