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Ostrovityanka [42]
3 years ago
15

A scientist counts 25 bacteria present in a culture and finds that the number of bacteria triples each hour. The function y = 25

∙ 3x models the number of bacteria after x hours. Estimate when there will be about 1170 bacteria in the culture
Mathematics
2 answers:
snow_tiger [21]3 years ago
8 0

the answer is 2.5 hours

Ksju [112]3 years ago
6 0
y = 25\cdot~3^x\\ \\ 1170 = 25\cdot~3^x\\ \\ \frac{1170}{25} = 3^x\\ \\ \frac{243}{5} = 3^x\\ \\ log\ \frac{243}{5} = log~3^x\\ \\ log~243-log~5=x~log~3\\ \\ 
 log~3^5-log~5=x~log~3\\ \\ 5~log~3-log~5 = x~log~3\\ \\ 5(0,48)-0,70= 0,48x\\ \\ 2,4-0,7 = 0,48x\\ \\ 1,7=0,48x\\ \\ x=\frac{1,7}{0,48}\\ \\ \large{\boxed{x=3,54\ h}}
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One nice thing about this situation is that you’ve been given everything in the same base. To review a little on the laws of exponents, when you have two exponents with the same base being:

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– Divided: Subtract their exponents

We can see that in both the numerator and denominator we have exponents *multiplied* together, and the product in the numerator is being *divided* by the product in the detonator, so that translates to *summing the exponents on the top and bottom and then finding their difference*. Let’s throw away the twos for a moment and just focus on the exponents. We have

[11/2 + (-7) + (-5)] - [3 + 1/2 + (-10)]

For convenience’s sake, I’m going to turn 11/2 into the mixed number 5 1/2. Summing the terms in the first brackets gives us

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And summing the terms in the second:

3 + 1/2 + (-10) = 3 1/2 + (-10) = -6 1/2

Putting those both into our first question gives us -6 1/2 - (-6 1/2), which is 0, since any number minus itself gives us 0.

Now we can bring the 2 back into the mix. The 0 we found is the exponent the 2 is being raised to, so our answer is

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