Answer :
Answer:
Number of bacteria at 11:53 for 53 min = [tex]2^{53}[/tex] = 9.007199 × [tex]10^{15}[/tex]
fraction that is full = [tex]2^{-7}[/tex]
Step-by-step explanation:
solution
Number of bacteria is express as here
Number of bacteria = [tex]2^t[/tex] ............1
so
bacteria at 1 min , 2 min and 3 min are
Number of bacteria at 1 min = [tex]2^1[/tex] = 2
Number of bacteria at 3 min = [tex]2^2[/tex] = 4
Number of bacteria at 3 min = [tex]2^3[/tex] = 8
and
as that Number of bacteria at 11:53 is for 53 min is
Number of bacteria at 11:53 for 53 min = [tex]2^{53}[/tex] = 9.007199 × [tex]10^{15}[/tex]
and
bottle is full at 12:00
so Number of bacteria at 12:00 for 60 min = [tex]2^{60}[/tex]
so fraction that is full = [tex]\frac{2^{53}}{2^{60}}[/tex]
fraction that is full = [tex]2^{-7}[/tex]
A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant.
Number of bacteria present in bottle at 11:53 are [tex]2^{53}[/tex].
At that time, fraction [tex]2^{-7}[/tex] or 0.00781 of bottle is full.
Since, single bacterium is placed in a bottle at 11:00am. It grows and at 11:01 divides into two bacteria. These two bacteria each grow and at 11:02 divide into four bacteria. Which grow and at 11:03 divide into eight bacteria.
So, a sequence is obtained 1, 2, 4, 8, 16, 32, 64, .........
We can also write as,
At 11:00 am, [tex]2^0[/tex] bacteria
At 11:01 am, [tex]2^1[/tex] bacteria
At 11:02 am, [tex]2^2[/tex] bacteria
Similarly, At 11:53 am, Number of bacteria present in bottle are [tex]2^{53}[/tex].
When the bottle will be full, that time number pf bacteria will be [tex]2^{60}[/tex].
S0, fraction of bottle is full at 11:53 is [tex]2^{53} /2^{60} = 2^{-7} = 0.00781[/tex]
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