Question

1. (l3) lake michigan is contaminated with bacteria. initially, the lake contained 4 square feet of bacteria. this area is growing by a factor of 3 every hour. assume that the relationship between hours and square feet of bacteria is continuous.

x (hours)	0	1	2	3
f(x) (# of bacteria)	4	12	36	108

a) write an explicit function for the relationship.
b) how much bacteria will be there after 5 hours?

Explanation:

Step1: Identify the form of exponential - growth function

The general form of an exponential - growth function is $f(x)=a\cdot b^{x}$, where $a$ is the initial value and $b$ is the growth factor. Given that the initial amount of bacteria $a = 4$ (when $x = 0$, $f(0)=4$) and the growth factor $b = 3$ (grows by a factor of 3 every hour). So the function is $f(x)=4\cdot3^{x}$.

Step2: Calculate the amount of bacteria after 5 hours

Substitute $x = 5$ into the function $f(x)=4\cdot3^{x}$. We get $f(5)=4\times3^{5}$. First, calculate $3^{5}=3\times3\times3\times3\times3 = 243$. Then $4\times243 = 972$.

Answer:

a) $f(x)=4\cdot3^{x}$
b) 972 square feet of bacteria