Question

3. Exponential Function:

Bacteria grows on food by doubling every hour.
If the bacteria initially covers 10% of the food, it will
cover the entire food source in 3.3 hours.

Answer 1

`y=0.1(2)^x`

where x is time in hours, and y is the number of bacteria in percent in decimal form.

Explanation:

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.
  If b > 1 then it is an increasing function.
  If 0 < b < 1 then it is a decreasing function.
• x is the independent variable.
• y is the dependent variable.

Define the variables:

• Let x = time (in hours)
• Let y = number of bacteria (in percent in decimal form)

If the bacteria grows on food by doubling every hour then the growth rate b = 2.

If the bacteria initially covers 10% of the food then y = 10% when x = 0.

Therefore, a = 10% = 0.1.

Therefore, the equation is:

`y=0.1(2)^x`

Substitute x = 3.3 into the found equation:

`\begin{aligned}x=3.3 \implies y & =0.1(2)^(3.3)\\& =0.9849155307...\\& =1.0\:\sf (nearest\:tenth)\end{aligned}`

As 1.0 = 100%, this justifies the claim.

`\begin{array}c\cline{1-3} x & y & y \\\cline{1-3} 0 & 10\% & 0.1\\\cline{1-3} 1 & 20\% & 0.2\\\cline{1-3} 2 & 40\% & 0.4\\\cline{1-3} 3 & 80\%\ & 0.8\\\cline{1-3} 4 & 160\% & 1.6\\\cline{1-3}\end{array}`