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The population of a wildlife habitat is modeled by the equation $P(t)=\frac{360}{1+6.2 e^{-0.35 t}}$ , where $t$ is given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity?

Board question 1 89. So in this case we have to find the ears when the percentage of home that we see ours is greater than 70. So we have to find when the pity We're talking about 189. When Petey is greater than 70, it means that 80 over one plus 63 times he raised to the power negative 0.63 T. Should be greater than 70. This means that the Syrian zeros cancer. If you simplify this, uh We are getting dividing both sides by eight, we can get one or one plus 63 days to the bone is 0.63. T. Is better than 7/8. Uh This can be read it and as we take the reciprocal of both sides, definitely the sign is gonna in ward. So that will be less than 8/7. And if you subtract subtract one both sides, so that's going to be 63 times the race to the pound. 0.63 He should be less than 1/70. Uh If we divide both sides by 63 so that's going to be raised to the bone negative 0.63 T. And seven entries 21 to 76 42 43 44. So that's gonna be a 44 one over here. And if I take the log natural log of both sides, that's gonna be uh negative 0.63 D. Is less than natural log of 1/4 41 Since we have a negative sign both sides. So let's multiply minus one. So the inequality will interchange. That's going to be Ln 441 So finally there's going to be equal to t is greater than 100. Natural log of 44 1/63. But this is the number of years after 1980. But the question is interested to find those ears In which this thing over. So those years will be, we are 1980 with these much years to get the actual ear in which this situation happens. So option B would be the correct choice. Thank you.

So, uh, in this case, we're going to use the natural growth model. So the formula for that is p of T equals peanut times e to the rt where yeah, p of t is the and population p not is the starting population are is the rate of growth. We're decline and T is the time. In this case, if we're going to predict how long it takes for the population to double, we need to know the rate at which the population is growing. To do that, we can use the example given in this case, the population of the otters went from 500 to 604 years. So that means, in our case, we have pe four equals 600 when p not was 500. So this means we have values for three of the four variables and so therefore we can get a value for are so we substitute these values into the formats of 600 equals 500 times e to the R times four and we solve for R. So divide both sides by 500. So that's 6/5 equals e to the four are then we can use laws of logarithms to rewrite this exponential equation as a natural algorithm. So the natural log of 6/5 equals four R and then divide both sides by four. So we get the natural log of 6/5 over. Four eagles are so now. If we want to look at how long it takes the population to double, we can use this value for our and we want our population to double so we could use the same starting population of 500. Or if we choose, we could just let the original population B one, which means our P value it would be, too, because it's going to double. And so we plug in those values and this time we're going to solve for T. So we've got two equals one time, see to the natural log of 6/5 over four times t. So that's one times Anything is that other objects it would just be too equals e to the natural log of 6/5 over 14. Use rules of algorithms to once again change this into a natural log equation. So the natural log of two equals the natural log of 6/5 over four times t multiplied by the both sides by the reciprocal mhm. And so we get t equals four times the natural log of two over the natural log of 6/5. And we can use a calculator to get an approximate answer for this problem. So when I put this into my calculator, I get four times. The natural log of two, divided by the natural log of six ifs, is approximately 15.2 years.

36. We have, hey equals P me far t We're trying to find the amount in the future. If we invest 25,000 that is growing interest continuously and a great, uh, my nose. It's five for 100 years, so that gives me a equals 25,000. Hey, to the 6.5 power E to the 6.5 power is 600 65. Wait one for two times 25,000 which becomes 16 1,000,000 628 1005 140 and 83 cents. And that's a dollar's. That would be my unit.

Base of her party. They were given that L. A is equal to 80. So it's all for T given l so we can plug that into our corrosion. So that's 87 minus Haiti over 63. And now let's focus into our Twitter. Okay, so this gives us a 5.646 nine a noun for B. We want to find a range pretty. So let's plug r TN you want to You evaluated at 24. So this is a native two points 57 natural log of 87 minus 24 over 63. Let's put that into our, um they're cuter. 87 minus 24. Divided by 63. That gives us a carol and then we won't t evaluated at seven. That's negative. 2.57 natural log of 87/87 over a 60 drink. So this would give us a demeanor. That's because we're getting really close to, um, our X axis beings are all okay, so we see that this is our domain from 24 to 7. Because when we have 87 we get the natural log is undefined at zero, and this year gives us are starting value at girl