Answer:
Part a) The number is 8
Part b) The initial number of bacteria is 4
Part c) [tex]y=4(8)^x[/tex]
Part d) [tex]y=100(8)^x[/tex]
Step-by-step explanation:
we know that
The equation of a exponential function is equal to
[tex]y=a(b)^x[/tex]
where
y is the number of bacteria
x is the time in hours
b is the base of the exponential function
a is the initial number of bacteria
Part a) When comparing the "# of Bacteria" each hour, what is the number being multiplied by each time?
we know that
For x=1 h ----> y=32 bacteria
For x=2 h ----> y=256 bacteria
For x=3 h ----> y=2,048 bacteria
For x=4 h ----> y=16,384 bacteria
For x=5 h ----> y=131,072 bacteria
For x=6 h ----> y=1,048,576 bacteria
so
256/32=8
2,048/256=8
16,384\2,048=8
131,072/16,384=8
1,048,576\131,072=8
so
the base of the exponential function b is 8
Part b) What is the initial number of bacteria? (at time zero)
we know that
The number of bacteria at time x=1 hour , divided by the number of bacteria at time x=0 must be equal to 8 (see part a)
so
[tex]\frac{32}{a}=8[/tex]
solve for a
[tex]a=32/8=4\ bacteria[/tex]
Part c) Write a rule for this table
we have
[tex]y=a(b)^x[/tex]
we have
[tex]a=4\\b=8[/tex]
substitute
[tex]y=4(8)^x[/tex]
Part d) Suppose you started with 100 bacteria, but they still grew by the same growth factor. Write the function rule for this situation
In this case
[tex]a=100[/tex]
[tex]b=8[/tex] ---> the growth factor is the same
so
[tex]y=100(8)^x[/tex]
