b Use the model you found in Part A to find the number of cells in 4 hours.
A
a y=6000(2.16025)^x
B
b 130 667
Practice makes perfect
a To find a formula to model our data, we need to consider the formula for an exponential function.
y= a b^x, where b>0
Here y represents the dependent variable, a stands for the initial amount, b is the growth factor, and x is the independent variable. At the beginning of the experiment, the time is 0 hours and there are 6000 bacteria cells.
When x=2, the number of bacteria cells is 28 000. We can use this information to assign values to the variables.
a= 6000
x= 2
y= 28 000
We are left unknown with the variable b, which we will solve for by substituting all our known values into the function.
y= a b^x ⇔ 28 000= 6000 b^2
We will now take our function and simplify it to solve for our growth factor, b.
By simplifying our equation, we find that b≈ 2.16025. We will substitute this value, along with our value of a, to find a function that models the number of bacteria after x hours.
y= a b^x ⇔ y= 6000( 2.16025)^x
The function that models the number of bacteria after x hours is y=6000(2.16025)^x.
b We can use our model to find the number of bacteria cells that can be expected after 4 hours by substituting x= 4 into our function.
y=6000(2.16025)^x ⇔ y=6000(2.16025)^4
By simplifying our function, we find that the expected number of bacteria cells is y ≈ 130 667.
