Exponential Growth : Advanced Problem

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dumluck: Junior | Next Rank: 30 Posts; Posts: 14; Joined: Wed Jan 12, 2011 4:19 am

by dumluck » Fri Apr 01, 2011 8:29 am

From Manhatten GMAT..

A strain of bacteria multiplies such that the ratio of it's population in any two consecutive minutes is constant. If the bacteria grows from a population of 5 million to a population of 40 million in one hour. By what factor does the population increase every 10mins?

I'm not sure how the above relates to the formula or indeed how you would solve it using this formula.
Final Amount = original amount * multiplier^(number of changes).

Another gripe I have with the Manhatten explanation is that they give the formula for exponential growth as y(t) = y_0.k^t but in the explanation for the answer they give it as y = k(R^t).

Any help would be gratefully appreciated.

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Source: Beat The GMAT — Problem Solving | Fri Apr 01, 2011 8:29 am

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by GMATGuruNY » Fri Apr 01, 2011 8:55 am

dumluck wrote:From Manhatten GMAT..

A strain of bacteria multiplies such that the ratio of it's population in any two consecutive minutes is constant. If the bacteria grows from a population of 5 million to a population of 40 million in one hour. By what factor does the population increase every 10mins?

I'm not sure how the above relates to the formula or indeed how you would solve it using this formula.
Final Amount = original amount * multiplier^(number of changes).

Another gripe I have with the Manhatten explanation is that they give the formula for exponential growth as y(t) = y_0.k^t but in the explanation for the answer they give it as y = k(R^t).

Any help would be gratefully appreciated.

The easiest approach -- by far -- would be to plug in the answer choices, which an actual GMAT question would include.
The answer choices would represent the factor by the population increases every 10 minutes.
Over 60 minutes, the population will be multiplied by the given factor 60/10 = 6 times.

Answer choice: âˆš2.
5,000,000 * (âˆš2)^6 = 5,000,000 * 8 = 40,000,000.

The correct answer is [spoiler]âˆš2[/spoiler].

We also could use the formula for exponential growth:

Final Amount = Original Amount * (Multiplier)^(Number of changes)

In the problem above:
Final Amount = 40,000,000
Original Amount = 5,000,000
Multiplier = x
Number of changes = 6 (We're looking for the change over 10 minutes, and this change will occur 6 times over the course of 60 minutes.)

Plugging these values into the formula:
40,000,000 = 5,000,000 * x^6
8 = x^6
x = 8^(1/6) = âˆš2.

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by GMATGuruNY » Wed Dec 12, 2012 8:18 am

dumluck wrote:From Manhatten GMAT..

A strain of bacteria multiplies such that the ratio of it's population in any two consecutive minutes is constant. If the bacteria grows from a population of 5 million to a population of 40 million in one hour. By what factor does the population increase every 10mins?

I received a PM asking me to explain the portion in red below:

We also could use the formula for exponential growth:

Final Amount = Original Amount * (Multiplier)^(Number of changes)

In the problem above:
Final Amount = 40,000,000
Original Amount = 5,000,000
Multiplier = x
Number of changes = 6 (We're looking for the change over 10 minutes, and this change will occur 6 times over the course of 60 minutes.)

Plugging these values into the formula:
40,000,000 = 5,000,000 * x^6
8 = x^6
x = 8^(1/6) = âˆš2.

When an exponent is raised to another power, MULTIPLY THE EXPONENTS:
(x^a)^b = x^(ab).

To simplify an equation with an exponent, raise each side to the RECIPROCAL POWER.
Given x^a = k, where k is a constant:
(x^a)^(1/a) = k^(1/a)

x^(a * 1/a) = k^(1/a)

x = k^(1/a).

In the problem above, since xâ�¶ = 8, we get:
(xâ�¶)^1/6 = 8^(1/6)

x^(6 * 1/6) = (2Â³)^(1/6)

x = 2^(3 * 1/6)

x = 2^(1/2)

x = âˆš2.

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