<PRE><B>Question 121704</B>
At time = 0 there are 24 spores.
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8 minutes later the number of spores has tripled. So after 8 minutes the number of spores is 24*3.
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8 minutes after that [16 minutes total] the number of spores again triples. You started this
8 minute block with 24*3 spores and you triple that to get [24*3]*3 = 24*3^2
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8 minutes after that [24 minutes total] the number of spores again triples. You started this
8 minute block with 24*3^2 spores and you triple that to get [24*3^2]*3 = 24*3^3
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8 minutes after that [32 minutes total] the number of spores again triples. You started this
8 minute block with 24*3^3 spores and you triple that to get [24*3^3]*3 = 24*3^4
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By now you may see that the pattern for this problem is 24*3^N where N is the number of the
8 minute periods you have gone through. Since you are to go through 60 minutes, if you divide
60 by 8 you find that you are going to go through 7.5 periods of 8 minutes each. So for this
problem the pattern will result in:
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Number of spores after 60 minutes = 24*(3^7.5)
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You can calculate 3^7.5 several ways. One is to use a calculator that has an x^y key and replace
x with 3 and y with 7.5 to find the answer of 3787.995116. Another way is to recognize that 3^7.5
is equal to (3^3)*(3^3)*(3^1)*(3^0.5). But 3^3 = 3*3*3 = 27 and 3^1 = 3 you can substitute
27 for 3^3 and 3 for 3^1 to get:
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(3^3)*(3^3)*(3^1)*(3^0.5)= (27)*(27)*(3)*(3^0.5) = 2187*(3^0.5)
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But 3^0.5 is 3^(1/2) and this is another way of writing the square root of 3. So 3^7.5 is
equivalent to 2187 times the square root of 3. And if you calculate this you again get that
3^7.5 = 3787.995116
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Now all you have to do to get the number of spores after 60 minutes is to return to the equation:
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Number of spores after 60 minutes = 24*(3^7.5)
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and substitute 3787.995116 for 3^7.5 and the equation becomes:
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Number of spores after 60 minutes = 24*(3^7.5) = 24*3787.995116 = 90911.88279 
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And this rounds to 90912 spores.
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Hope this helps you to understand the problem a little better and to see how you can think
your way through to the answer (with a little calculator help).
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