Answer:
[tex]y=0.1(2)^x[/tex]
where x is time in hours, and y is the number of bacteria in percent in decimal form.
Step-by-step explanation:
General form of an exponential function:
[tex]y=ab^x[/tex]
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:
[tex]y=0.1(2)^x[/tex]
Substitute x = 3.3 into the found equation:
[tex]\begin{aligned}x=3.3 \implies y & =0.1(2)^{3.3}\\& =0.9849155307...\\& =1.0\:\sf (nearest\:tenth)\end{aligned}[/tex]
As 1.0 = 100%, this justifies the claim.
[tex]\begin{array}{|c|c|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}[/tex]
