Second Expression: y=1(2)^(3x) Covered Area: About 0.707 square centimeters
Practice makes perfect
a The area of the petri dish covered by bacteria increases at an exponential rate. This means we must model the area with an exponential function.
y=C(a)^x
We also are told that the area doubles every 20 minutes and starts with 1 square centimeter. This means C= 1 and a= 2. So far, all of the models have the same parameters.
y= 1( 2)^x
Now we can analyze the expressions from each person.
Bailey
Since the area doubles every 20 minutes and there are three 20 minute intervals in an hour, the exponent should be 2* 3 = 6 if x describes the number of 20-minute intervals during the two hours. This matches how Bailey has reasoned.
Carmen
In Carmen's expression we have two exponents, one that is 1/20 and another that is 120. There are 120 minutes in two hours and the area doubles every 20 minutes. If we simplify the expression we would get the same exponent as Bailey.
Demetri
In Demetri's expression we also have two exponents, one that 3 and another which is 2. The first exponent describes the number of times the bacteria doubles in an hour. The second exponent is the number of hours.
Covered Area
Let's evaluate one of the expressions.
1(2)^6=64 cm^2
The area covered by the bacteria after 2 hours is 64cm^2.
b We know that the area covered by bacteria can be described by the following exponential function.
y=1(2)^x
In this equation x is the number of 20-minute intervals. To express x in another time unit we have to manipulate the exponent. For example, if we divide x by 20 we get x in minutes.
x in minutes: y=1(2)^(x/20)
We can also let x represent hours if we multiply x by 3.
x in hours: y=1(2)^(3x)
To determine the area covered 10 minutes before they started the experiment, we will substitute x=-10 into the equation where x is described in minutes.
