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An aircrah (at Z) Is spolted by IwO observers (at X and Y) who ate L = 1850 (oet apet. As the aiplane passes aver Iha Iino jolning Iham; oach obearver takos a elghl...

Question

An aircrah (at Z) Is spolted by IwO observers (at X and Y) who ate L = 1850 (oet apet. As the aiplane passes aver Iha Iino jolning Iham; oach obearver takos a elghling 0f Iho anala of olavalion Iha plano. u8 indicoted liquro |A=35", and B = 309 how high Ile airplano?Tha olevallon ol (ho plane (s approxlmaluly (oul (Do not rourd unlil Ine final answar Thon round Io Iwo decimal places 05 neoded )

An aircrah (at Z) Is spolted by IwO observers (at X and Y) who ate L = 1850 (oet apet. As the aiplane passes aver Iha Iino jolning Iham; oach obearver takos a elghling 0f Iho anala of olavalion Iha plano. u8 indicoted liquro |A=35", and B = 309 how high Ile airplano? Tha olevallon ol (ho plane (s approxlmaluly (oul (Do not rourd unlil Ine final answar Thon round Io Iwo decimal places 05 neoded )



Answers

Near a buoy, the depth of a lake at the point with coordi-
nates $(x, y)$ is $z=200+0.02 x^{2}-0.001 y^{3},$ where $x, y,$ and
$z$ are measured in meters. A fisherman in a small boat starts
at the point $(80,60)$ and moves toward the buoy, which is
located at $(0,0) .$ Is the water under the boat getting deeper
or shallower when he departs? Explain.

In this problem were given these shown information with a equation Z that represents the depth of the lake and told that a fisherman starts at a 0.80 60 and is moving towards the 600.0 and were asked to find out of the water is getting deeper or shallower. The way we want to do this is to find the directional derivative. And if that derivative is positive, the water is getting deeper because the derivative of the function in the direction that the fishermen is traveling, it's positive, meaning that the death is getting larger. Which means that the vote was getting the water under the boat was getting deeper and vice versa. If it's shall getting shallower. And so to Father directional derivative, we're going to take the partial of Z with respect to X, and when we do this we get 0.2 times two x, which is 20.4 Thanks, then the partial with respect to why is going to be negative 0.1 times three y squared or negative 30.3 Why cute and we're going to evaluate this at the 0.80 60 and when we do this, we get the following point. Oh, for times 80 is equal to 3.2. And then when we do evaluate the partial for why at the point, we get 60 square, just 3600 times negative 0.3 and we get negative 10.8. And so that is our Grady in Vector 3.2. Common negative. 10.8. And now we want to find the directional doctor we're going to do. This is going We're going to take the endpoint, which is 00 and subtract er starting 00.80 60. And so we're going to take 00 and subtract any 60 and get the resulting vector negativity. Negative. 60. No, I wouldn't want to make this a unit vector by dividing it by the square root of the sum of the squares of each component. So negative 80 squared plus negative 60 square, and that is going to be equal to 80 times 80 is 6400 60 times 60 is 3600. And so that's 10,000. And the square root of 10,000 is 100. And so I'm going to distribute that that into each term and get negative. 80 over 100 and notice 60 over 100. And we can simplify these to get the factor. Negative. Four over five and 3/5, and now we're going to dot product. This with the specter of here that we found for the grating factor I would actually change for negative forfeits and 3/5 into decimals to make our lives a little easier. Negative forfeits. His negative 0.8 3/5 is 0.6, and now when we do this math, we get 3.2 times negative 0.8, and that is equal to negative 2.56 and then we get negative 10.8 times negative. Excuse me down here. That should have been a negative times. Negative point sits, and that is equal to positive 6.48 And so if you take 6.48 and subtract 2.56 we get 3.92 which the exact number does not exactly matter, but because it's positive, we know that the water is getting deeper

In this problem. We're even this figure here. This is not This is so. This is east. This is best. And this point is See, this point is a This is B. This is D. Now the distance between B and D is 4 26 ft and the angles here are to be 7.5 degrees on 12.3 degrees. Now we have to find that How far will others Hugo at a have good travel to reach any survivors at B Therefore, in Triangle BCB, engine C is equals toe beauty over B C, which is equals toe. BC is equals toe 4 26 over engine, 12.3 degrees. We get 4 26 over point to 18 That forced defined this you get this is it was toe 1954.1 to 8 ft. Now in triangle B, a C angle a CB music was toe 57.5 degrees plus 12.3 degrees uses. It was to 69.8 degrees. Now sign C is equals toe a B over BC that before we get a b is equals toe. BC maybe tie sign. See they get 1954.12 Big My Life sign, 69.8 degrees. Therefore defined this We get a B is equals toe 1833.90. Therefore, a rescue boat will have to travel dispense off one a 33.19 distance to the point to be so the finance solution is this one.

This question. We have this, uh, football game situation. So you go up there is kicks the ball and a resource be off for initial speed or 50 ft per second and go 60 degrees. Then a player start from another player. Start from be, uh, run along BC. Then he wants to catch the ball when he reaches point seat. Everyone to find this speed. Okay, so that he's able to catch the ball. Okay, So to do this, um, you are going to consider so this project how emotion involved. Okay, so this is project how emotion p Then this is just normal, uh, media motion. Oh, Okay. So for projectile motion for the football, he went to calculate the time of flight. Okay, at the time of flight. Alright, Beachy. Great. So, um, initially, uh, okay. I'm going to using. Uh huh. We b y eyes equal to we y initial. Plus our a y t. Right. So I'm going to take, um downward to be positive. S o t will be b y minus b y show divide by a y. Okay, So, um, this is why at point c would be so a c mhm Um Okay, so this is the me initial. So this is a very final right. So Okay. So initially we have negatively signed data, but at the same time was subject. And finally, we also have the side data. Okay, then the Rabbi G. Okay, so this is to read science data. Divide by G. You substitute the values. We is, uh, beach 50 in sign. 60 degrees G is 32.2. You calculate the time would be 2.69 seconds. Okay. And we want to find the distance a c a a c will be, uh, me cross eyed data times t k. So because the, uh, the football is traveling along the y axis, and the horizontal component of the velocity does not change its just beacause data multiplied by the time. So we have, uh, 15 co sign 60 degrees? Yeah. Times 2.69 In a couple of days, you get 56 10 Mhm. 67.2 ft. Uh huh. And so this this whole distance is 67.2 heat. Okay. Which means that this is ah, 47.2 ft can because there's a 20 ft over here. So distance off. Easy. Okay, so we will be using progressed. Um, you know that this is 30 shit, And then this is from here to here is 47.2. And so you have 30 square plus 47.2 square square It. This is our 56.0 ft and then the speed of the player. Okay. Okay. Um, instance, we see divide variety, a 56.0 sheets. You are by 2.69 seconds in calculating you get 20.8 feet per second. Okay, so this is the answer for this question, and that's all.

In this question, the football player a chose the football. Uh, because initial speed off, we go through 50 ft second. Then at the same time, player starts running from point B 23 ft per second. So we want to find out whether player B can reach point C before the ball reaches before the football reaches Point. See? Okay, so in this question way have a projectile motion, okay? And then this is just the emotion without exaggeration. Okay, So, uh, for the projectile motion, First we need to, uh, so we one thing we need to find a distance, a see, also need to find a distance. BC, and, uh, calculate the speed and time and, uh, and see if there be can reach point, See before the football. Richard's point. See? Okay, So, uh, let's the time of flight. I'll be he Okay, so we'll be using b y finance. It goes toe B Y initial. That's a Y T. Yeah, I'm going to take downward to be positive. Mhm. So, um, so if you look at and see the projectile motion, Okay, so we have e at a pointing like that. E s c pointing in this way. Okay, So the initial is, uh, going to be negatively side data and then finally be, uh, recent data. Okay, so we decided data because to the negative side data hice duty. So he isn't going to to be side data. Uh, the G. So you substituted numbers two times 50 sign, 60 degrees by 32.2. Calculate the tea. It is 2.69 seconds. Thanks. And, um, you can create a distance A C s. So this is just we not because I data times t okay, Because looking at this eyes moving along, uh, the Y axis. And so this component of velocity is we We're not consign 60 degrees, right? And then you just multiply the time you get 2.6 you get, you have 15 for sign, 60 degrees times 2.69 is the time that we follow. Just now, get 67.2 ft and then, uh okay, so it's back to all the diagram. So a c from here to hear 67.2 ft, which means that from here to here eyes 47.2 it because, uh, this part is 20 ft. So then we can find BC using product restaurant. He re square plus 47.2 square square roots. You get 56.0 ft. Okay, so, um, and we are given that given the speed off here, Yes, uh, 23 ft per seconds. Right. So and, uh, if we just calculate the minimum speed required, Okay, minimally required will just be 56. Divide by 2.69 when you calculate this to be 20.8 feet. Correct seconds. So the speed up here is greater than the minimum speed. The quiet. Okay. Which means that the player and reach he's already reached a football or rich Quincy before that. Football before the ball gets there. Okay. Okay. So this is how we decide. Uh, reached we? This is how we reached the conclusion. Okay. Based on is on some is on the calculation of speed. Okay, if by campaigners, we appear to the minimum speed. Okay, so that's how we solve this problem. And that's all


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