WEBVTT
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A population of bacteria decreases as a result of a chemical treatment.
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The population π‘ hours after the treatment was applied can be modeled by the function π of π‘, where π of π‘ is equal to 6000 times 0.4 to the power of π‘.
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So thatβs describes the scenario in the question.
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This is a multipart question; there are two parts.
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The first part is βwhat was the population when the chemical was first applied?β
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The second part is βwhat is the rate of population decrease?β
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Naturally, weβre going to answer the first part of the question first: what was the population when the chemical was first applied?
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Well, we have a function π of π‘, which tells us the population π‘ hours after the treatment was applied.
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So asking for the population when the chemical was first applied, the initial population, weβre asking what was the population at π‘ equals zero.
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That is π of zero, which is the population zero hours after the treatment was applied.
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And to find that, we just substitute into the expression we have for π of π‘.
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Substituting in zero, we get π of zero equals 6000 times 0.4 to the power of zero.
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Anything to the power of zero is one, and so π of zero is 6000.
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So the answer to the first part of the question βwhat was the population when the chemical was first supplied?β is π of zero or 6000.
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The next part of the question is βwhat is the rate of population decrease?β
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Again, we use the definition of the function π of π‘.
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Weβve already seen in the first part of the question that the initial population π of zero is 6000.
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π of one is the population after one hour.
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Substituting this into the formula we have for π of π‘, we get that π of one is 6000 times 0.4 to the power of one, which is of course just 6000 times 0.4 or 2400.
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The population of the two hours is π of two or 6000 times 0.4 to the power of two.
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Again, weβre just substituting in to the expression we have, and this is 960.
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The actual values arenβt actually important here.
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What is important is that the population at a certain time is 0.4 times the population one hour before.
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Another way of saying this is that the population is 40 percent of the population the hour before.
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And another way of saying this is that the population decreases by 60 percent every hour.
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So what is the rate of population decrease?
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It is 60 percent per hour.
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So looking at our function π of π‘, the multiplicative constant of 6000 tells us about the initial state β in our case the initial population.
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And the base 0.4 tells us about the change over time β in this case how much the- in this case how much the population decreases over time.