Bacteria are growing in a liquid growth medium. There were 300,000 cells per milliliter during an initial observation. The number

Question

Bacteria are growing in a liquid growth medium. There were $300,000$ cells per milliliter during an initial observation. The number of cells per milliliter doubles every $3$ hours. How many cells per milliliter will there be $15$ hours after the initial observation?

A.

$1,500,000$

B.

$2,400,000$

C.

$4,500,000$

D.

$9,600,000$

Accepted Answer

Choice D is correct. Let y represent the number of cells per milliliter x hours after the initial observation. Since the number of cells per milliliter doubles every 3 hours, the relationship between x and y can be represented by an exponential equation of the form y = a b x k , where a is the number of cells per milliliter during the initial observation and the number of cells per milliliter increases by a factor of b every k hours. It&rsquo;s given that there were 300,000 cells per milliliter during the initial observation. Therefore, a = 300,000 . It&rsquo;s also given that the number of cells per milliliter doubles, or increases by a factor of 2 , every 3 hours. Therefore, b = 2 and k = 3 . Substituting 300,000 for a , 2 for b , and 3 for k in the equation y = a b x k yields y = 300,000 2 x 3 . The number of cells per milliliter there will be 15 hours after the initial observation is the value of y in this equation when x = 15 . Substituting 15 for x in the equation y = 300,000 2 x 3 yields y = 300,000 2 15 3 , or y = 300,000 2 5 . This is equivalent to y = 300,000 32 , or y = 9,600,000 . Therefore, 15 hours after the initial observation, there will be 9,600,000 cells per milliliter. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

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