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During a 3h-experiment, the number of bacteria increased from 10^4 (at

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During a 3h-experiment, the number of bacteria increased from 10^4 (at  [#permalink]

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New post 26 Feb 2019, 15:48
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GMATH practice exercise (Quant Class 12)

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During a 3h-experiment, the number of bacteria increased from 10^4 (at the start) to 8 times this value (at the end), according to a biological law associated with an exponential function (as shown), where a and b are positive constants. If Madame Curie knows that a certain critical number of bacteria in this experiment is reached at exactly 30 minutes after the experiment begins, which of the following is closest to this critical value?

(A) 20,000 bacteria
(B) 18,000 bacteria
(C) 16,000 bacteria
(D) 14,000 bacteria
(E) 12,000 bacteria

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: During a 3h-experiment, the number of bacteria increased from 10^4 (at  [#permalink]

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New post 27 Feb 2019, 08:50
fskilnik wrote:
GMATH practice exercise (Quant Class 12)

Image

During a 3h-experiment, the number of bacteria increased from 10^4 (at the start) to 8 times this value (at the end), according to a biological law associated with an exponential function (as shown), where a and b are positive constants. If Madame Curie knows that a certain critical number of bacteria in this experiment is reached at exactly 30 minutes after the experiment begins, which of the following is closest to this critical value?

(A) 20,000 bacteria
(B) 18,000 bacteria
(C) 16,000 bacteria
(D) 14,000 bacteria
(E) 12,000 bacteria

\(f\left( t \right) = a \cdot {b^t}\,\,\,{\rm{bacteria}}\,\,\,\,\,\left( {{\rm{at}}\,\,t \ge 0\,\,{\rm{hours}}} \right)\)

\(? = f\left( {{1 \over 2}} \right) = a \cdot \sqrt b\)


\(\left( {0\,;\,{{10}^4}} \right) \in \,\,{\rm{graph}}\left( f \right)\,\,\,\, \Rightarrow \,\,\,\,{10^4} = a \cdot {b^0} = a\,\,\,\,\left( * \right)\)

\(\left( {3\,;\,8 \cdot {{10}^4}} \right) \in \,\,{\rm{graph}}\left( f \right)\,\,\,\,\, \Rightarrow \,\,\,\,8 \cdot \,{10^4}\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,{10^4} \cdot {b^3}\,\,\,\, \Rightarrow \,\,\,\,b = 2\)


\(?\,\, = \,\,{10^4} \cdot \sqrt 2 \,\, \cong \,\,1.41 \cdot {10^4} = 14100\,\,\,\,\, \Rightarrow \,\,\,\,\left( {\rm{D}} \right)\)


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: During a 3h-experiment, the number of bacteria increased from 10^4 (at   [#permalink] 27 Feb 2019, 08:50
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