WEBVTT
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Hey, it's clear. So when you read here
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Server for partying, we have dp over. DT
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. This is equal to keep p minus m.
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When we calculate for a d. T. We
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got a DP over. Keep he minus. Um
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, we're gonna substitute X kee p minus, um
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, and then four. And then we're gonna integrate
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both sides so we get one over K. The
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integral of K C P or her keep p minus
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M is equal to the integral d t we substitute
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this be we're going to get won over K.
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Shelton of X. It's equal to t. Yeah
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, so we only substitute back. You get one
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over K l and KP minus um is equal to
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t plus c. So to find the constant we're
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gonna substitute T is equal to zero and P is
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equal to p sub zero. So we get one
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over K l n. Okay. He subzero minus
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m is equal to see and we get one over
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k felt and of K P minus, um,
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this equal to t plus one over K felt and
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of K piece of zero minus m. We end
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up getting l on of KP subzero Linus. Um
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over K piece up zero minus m is equal to
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Katie. So we raised the power on the base
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e for both sides to get k p minus m
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over K p sub zero minus m is equal to
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e the k t. And we end up getting
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p of tea to be equal to piece of zero
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minus, um, over k times e the k
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t plus m over for part B. We're gonna
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use the equation we got from part, eh?
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Were given that dp over d t is equal to
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k T minus m. So using the equation,
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we're gonna substitute the value of P. Then we
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end up getting deep. He over DP is equal
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to K times piece of zero minus and over King
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B to the K T plus, um, over
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K minus. We get this to be equal to
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K piece of zero minus um times e Katie.
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You know that one dp over DT is positive than
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P increases asked. He increases and wanted negative.
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Then p decreases as tea increases. So we know
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that the population will grow exponentially if dp over d
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t is bigger than zero. This implies that this
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equation would as well be bigger than zero. So
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we get Hey, Humpy. Subzero minus M is
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bigger Zero. So we get the population will grow
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if m is less than K comes piece of zero
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. So we're gonna erase party now. And when
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we look at part see, we're going to use
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the equation Read arrived from a Let me reiterate that
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. So we get we got p of tea is
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equal to piece of zero minus m o over k
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times e to the k t plus em over King
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So we know that it's given that d p over
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d t is equal to k p minus m so
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we just substitute the values to get dp over.
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DT We did the same thing as we did here
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, which we ended up getting king He sub zero
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minus m times e to the k t. We
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said that the conditions when dp over DT is positive
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and negative. But when it zero, then we
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know that pee is gonna be constant. S t
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increases. We know that the population is constant when
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dp over d t is equal to zero. So
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this implies that this has to be equal to zero
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as well. So we get K piece of zero
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minus M is equal to zero and we end up
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getting m is equal to K comes piece of zero
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. So that's when the population will remain constant.
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We know that it declines if d pee over DT
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is less than thorough. So he used this equation
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and we just make this less than zero. So
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yet, kay piece of Serra minus m iss less
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than zero to get em is bigger than Cape piece
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of Zero. So that's when the population will decline
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me. Erase this part. So we're gonna look
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at the population now, uh, 18 47.
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So we know that the population was eight million in
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Ireland, so we just get the equation. P
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subzero is equal to eight times tend to the sick
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power. So we know that K is equal to
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the difference between birth and death. Rates over the
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population were given that the difference is 1.6%. So
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Kay has to be equal to 0.16 We know that
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there's 210,000 people, um, per emigrated from Ireland
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so M is equal to point to one times 10
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to the sixth power. So we just yet K
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times piece of zero is equal to 0.16 times eight
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times Tend to the sick when we get 0.1 to
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8 times turned to the sixth and we see that
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M is bigger than K times key subzero. So
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looking at part, see, we know that the
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population is declining.