Answer:
a) each hour the number is multiplied by 16. Such a geometric sequence is described by an exponential growth function.
b) n = 16^t . . . . . n is number of bacteria; t is hours
c) about 7.923×10^28
d) the rule would be multiplied by 100/16=6.25, so becomes n=6.25·16^t
Step-by-step explanation:
a) We presume the table represents exponential growth because it was formulated with that purpose in mind. ("Why?" is always a tricky question.) It represents exponential growth because each table entry is 16 times the one before. A constant multiplier like that is characteristic of exponential growth.
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b) In part (a) we noted the common ratio is 16. Since the first term (for t=1) is also 16, we can write the equation as ...
n = 16^t
where n is the number of bacteria and t is time in hours
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c) Evaluating the formula for t=24, we get ...
n = 16^24 ≈ 7.923×10^28
n = 79,228,162,514,264,337,593,543,950,336
This is the estimate of the number present after 24 hours based on the formula.
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d) If the starting number is different, the rule is multiplied by a factor that gives the appropriate starting value. The above answer assumes your "starting value" is 100 after 1 hour. If the starting value at 0 hours is 100, then the rule of part (b) gets multiplied by 100, not 6.25
n = 100·16^t . . . . . starting with 100 at t=0
n = 6.25·16^t . . . . starting with 100 at t=1
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Comment on exponential bacteria growth
The number of bacteria estimated in part (c) have an approximate volume of 102 cubic kilometers, roughly the volume of Lake Nicaragua.
