So the first thing we want to do with this is to determine the velocity for this. Well, if this is our position function T then to get the velocity, we just take the derivative of this with respect to time. So we'll take the derivative of this. So is what unplug essence. And the first thing I'm going to do is pull that 160 out just due to the constant rule. So this would be 1 60 D by D. T of 1 40 minus one plus e to the negative t to fourth. Now we can use the sum and difference rule to distribute this across. So 1 60 and then I also use the constant rules. You want to pull that 1/4? So be 1/4 D by D t t minus d by d t of one plus D by d t of e to the negative t to the fourth. And we could go ahead and take each of these derivatives by themselves. So first, the derivative of tea Well, that's just one the derivative of any constant. To include one is going to be zero and now to take the derivative of E to the negative t four we're going to need to use, um chain. So first it's right out what we have over here. So 14 times one gives us 1/4. Now Chain will remember says we take the role of the outside function, which is going to be e to the T. And so that's just gives us e to the T. And then we plug back in the negative t fourth. And then we take the derivative of what we have on the inside, which is t to the negative forces. And now that there again, we would just pull out the negative one pork, take the derivative of T as within this becomes negative 1/4. So let's go ahead and write all that up. So we have 1 60 oh, 1/4 plus or not, plus minus because it's negative. Negative. 1/4 1 60 he to the negative t fourth. I was just repeat that 1 60 just to make it look a little bit prettier because I don't like to have to look at fractional. This 1 60 is not there. I was getting ahead of myself. Yeah, I don't like to look at fractions if I don't have to So disturbing. That gives 40 minus 40 e to the negative t four. So this is going to be our velocity function based off of Thai. Now, the next thing they want us to do is to show so be they want us to show that our acceleration is going to be 10 minus 1/4 of our velocity. Eso first, let's just go ahead and plug it in over here toe, see what we get just so we can kind of like, check that this is true. So it will be 10 minus 1/4 and then we're taking this and plugging it in for V So 40 minus 40 e to the negative t over four and then we distribute the 1/4. So that gives us 10 minus 10 plus 10, 8 negative teeth. Fourth, which those tens cancel out with each other and we're going to just be left with 10 e to the negative t to the fourth. So this right here is what we want to show, So we haven't actually shown this yet. This is just what we're trying to get to. Alright so to actually get the acceleration the proper way, though, so acceleration is supposed to be the derivative with respect to time of our velocity function Our velocity function we found in the last step. So this will be d by D t of 40 minus 40 e to the negative t to the fourth. And now we can go ahead and distribute this So that gives D by d t of 40 minus So we can use thesis, um, and scaler rule for this over here on the right side, our difference in scaler. So it be 40 d by d t of e to the negative t fourth and again the derivative of 40 0. And then we already took the derivative e to the negative t fourth. But let's just go ahead and repeat those steps again. So there's going to be zero minus 40 so we use changeable. So first we just have e to the negative t forth by itself because derivative eaten X is just eat the X. Then we take the derivative of what was on the inside, which was that negative t fourth and again, This is negative 1/4. And if we multiply everything together. So negative 40 times negative. 1/4 is going to be 10 e to the negative t fourth, so you can see these two are the same. So it checks out now the next thing they want us to do is to find the terminal velocity, which they tell us is just limit as our velocity goes to infinity. So let me look at what our velocity was again. So it's 40 minus that. Okay, so down here we want to do the limit as t approaches infinity of the which again was limit as t approaches Infinity Uh, e or 40 minus already e to the negative t fourth. So if we were to just plug this limit indirectly, this is going to give us a 40 minus 40 e to the negative infinity over, for which that's just going to be negative infinity and then e to the negative infinity. Well, let's go ahead and reciprocate that. So it be 40 minus 40 over e to the infinity and then eats of infinity is infinity. So it be some constant over something going to infinity, which is going to be zero. So this is zero, which means our terminal velocity, which they had losses VR is going to be 40. And actually, do they tell us what units this is in? Um, so it's in meters and our time is in seconds, so actually this is going to be in meters per second. So this is the terminal velocity that we end up with. And then lastly, they want us to determine at what time do we get 95% of the terminal velocity? So that means we're going to set the velocity equal to 90% of this. So down here d time when velocity is equal to 0.9 five VR. So we will do 0.95 times 40 and then we set this equal to 40 minus 40 e to the negative t over four. Now, before I actually simplify this 40 over here, I'm just going to divide everything by 40 because they'll just make it look a little bit nicer. And since we have a 40 everywhere, it was just kind of all cancel out like that. And that's going to give 0.95 is ableto one minus e to the negative t to the fourth so we can go ahead and subtract one over. So it gives us negative 0.5 is equal to negative e to the negative T or four. Then we can multiply each side by negative. So the 0.5 and e to the negative teach fourth. Then we could take the natural log on each side. Uh, any log will do, but natural law will be the most natural one to use. And then that gives natural log of 0.5 is equal tube negative t over four, and then we can multiply over by that negative four. So we get t is equal to negative four natural log of 0.5 Right? And now that we have this time well, actually, let's just see what this is because I mean, I don't even know what that number means. If I were to look at it so natural, log a 0.5 times. Negative four. So this is approximately 11.98 seconds. So we have this here and then the last thing they wanted us to do was to figure out how far we have fallen so Let's go ahead and take this and then plug it in to our very first equation. Um, at the very top here. So let me come back up and pick this up and scoot it all the way down. And then we will need to plug in that value we just got into here. So now to get the distance, like I was saying, we just take this number, plug it in for these teas here. Alright, So s of negative four. Natural log of 0.5 is going to be so it's 1 60 1/4 actually, the 1/4 and that negative or just cancel out. So this, but actually just become negative. Natural log of 0.5 The minus one stays there. And then when we plug that in here, so it would be e raised to the so the negative one in four again counts out. So that's just gonna get us national log of 0.5 and the natural log or eat to the natural log. Cancel that, and that's just 0.5 So this is going to be 160 minus natural log of 0.5 minus one plus 0.5 And then we can combine those to be, um, 1 60 negative natural log of 0.5 minus 0.95 and I'll go ahead and pull the negatives out just to make it look a little bit prettier. So negative 1 60 natural log of 0.5 plus 0.95 And if we were to go ahead and plug this into a calculator because, I mean, honestly, I don't even know what that number is even close to so nuts log a 0.5 loss 0.4 point 95 is going to be about negative to that. We multiply that by negative 1 60 and so this is going to give us something approximately equal to 327.3 meters. So this is supposed to be our how much we fell after we've reached 90 95% of the terminal lawsuit. And if you used a slightly different number like if you would have rounded as opposed to keeping exact um, you might get something slightly different here, but as long as it's within a couple of meters, I would say it's a good chance that you plugged everything incorrectly.