Tracking a dive A biologist standing at the bottom of an 80-fo...

Question

Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure).

b. What is the rate of change of θ with respect to the bird’s height when it is 60 ft above the ground?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A woman stands on a flat ground surface at the base of a 90 ft tall observation tower. A hawk takes off from the top of the tower, descending at an angle of 45 degrees below the horizontal. Determine the rate at which the angle of elevation theta is changing with respect to the height of the hawk when it is located 65 ft above the ground. Awesome. So it appears for this particular prom we're ultimately trying to figure out what the rate at which this angle of elevation data is changing with respect to the height of the hawk when it is located 65 ft above the ground. So that once again, that's what we're ultimately trying to solve our, we're trying to figure out this rate at which the angle of elevation theta is changing with respect to the height of the hawk. So with that said, before we start solving this particular problem, let's take a moment to look at our figure that is provided to us by the prom itself. As you can see, we have our observation tower represented by this blue building, and as you can see from the flat ground surface going up to the top of the tower is 90 ft. We see our woman wearing a blue dress standing on the flat ground next to our observation tower. And as you can see, she can look up and she could see this hawk, which is descending at a diagonal line. As we can see, we have this dotted diagonal line that denotes our flight path, and as you can see, it takes off from the top of the observation tower and it's flying diagonally down towards the line of vision of the woman and the hawks denoted by this black shape. And as you can see, the angle of It's of the descent, so the angle of descent for this hawk is 45 degrees below the horizontal as indicated in our figure. Now as you can see, we have some distance X going out to where the hawk is. So like where the woman's standing out to where the hawk is located, it's going to be some distance, whereas there's also gonna be another value in which we could say this is some distance X, and we could say from where the hawk is located up in the air above the ground, it's gonna be some height value H, which we do not know. Awesome. So now that we have a good visualization of what's happening in this particular prompt, let's make this figure a little bit more simple for us to use for our solving methods. So let's create more like a skeleton version of this figure. That's provided to us so we can better visualize what's happening. So, as you could see, they have like they give us a more accurate, like, overall picture of what's happening. We could see we have the height of our tower total, like the observation tower is 90 ft, whereas if we were to break up the little like the diagonal flight path into a tiny little triangle shape as you could see, so this prom. In order to help us solve it more effectively, it's going to be very important that we have an understanding of geometry and trigonometry to understand like how triangles relate to like, how we, you know, we can relate triangles to our solving methods. So that's very important to help us solve. So as you can see, when we consider the flight path of our hawk to be triangular in shape, we could see. That we have the angle of descent, which is 45 degrees, and the top, so if we were to make it into a little triangle here, we could see that it's gonna be 90 minus H, which we'll discuss more in a minute why it's 90 minus H. And then we could say, like we said, if we had some distance X from where the woman's standing at the base of the tower going out to where the hawk is descending, that's at some distance H above the ground. And then we have these two red parallel lines which denotes that these two legs of the triangle are the same length. And because we know that the same length we have are 45 degree angle, so it's gonna be 45, 45, 90, we have a 90, like a right triangle, and all triangles add up to 180 degrees on the inside. So that's really helpful to note. So with that in mind, first off, in order to solve for this particular problem, we need to figure out an expression for the angle of like elevation theta in terms of age. We need to recall a note once again that the hawk takes off from the top of this 90 ft tower and it descends at a 45 degree angle below the horizontal. And as a result, this will mean that the hawk's flight path forms an isosceles right triangle. Where one leg is the vertical distance descended, which is 90 minus H, and the other leg is the horizontal distance from the tower X, as described in our skeleton diagram. So, since the angle of descent is 45 degrees, the isosceles property, as we should recall, tells us that the horizontal distance will be equal to the vertical distance, meaning that we could safely say that X is going to be equal to 90 minus H. So the woman, the hawk, and the point on the ground directly below the hawk will form a right triangle where the height of the hawk above the ground is H on the opposite side. And without a doubt we can also state that the horizontal distance between the woman which is located or who is located at the base of the tower and the point directly below the hawk will also be. X is equal to 90 minus H, which is on the adjacent side. So now we need to express theta in terms of H. So we need to recall and use the tangent function for the angle of elevation theta. So as we should recall, we need to note that tangent of theta is equal to the opposite side divided by the adjacent side, referring to the legs on our triangle. Which will be equal to H divided by 90 minus H. So when we rearrange this equation to isolate and solve for theta all by itself, we will find that theta is equal to inverse tangent of H divided by 90 minus H. Awesome. So now our next step is we need to determine D theta by DH. So as we should recall, D by DH is equal to or rather D by DH, so D by DH. So this is a fancy way of saying we're taking the derivative with respect to H, so D by DH of. Inverse tangent of H. is going to be equal to 1 divided by 1 plus H2. So now, let's scroll down so we have a little bit more room. So now our next step is we need to recall and use the chain rule to help us solve for D theta by DH. So as we should recall, using the chain rule, D theta by DH is going to be equal to One divided by one plus. H divided by 90 minus H. And then we're gonna have to take H divided by 90 minus H, and we're gonna take this to the power of 2. So, it's gonna be 1 divided by 1 plus H divided by 90 minus H to the power of 2. And then we need to multiply this expression by 90 minus H multiplied by 1 minus H multiplied by -1, all divided by 90 minus H2, fantastic. So now what we need to do is we need to take this big old long expression and we need to simplify it down to its most simple form, which I'm going to skip a few steps, but when we do that, when it's in its most simple form, we will find that D theta by DH and once again we're just using algebra to move things around and put it in its most simple form, and when we do, we will find that D theta by DH. It's gonna be equal to 90 divided by 90 minus H to the power of 2. Plus H to the power of 2. Fantastic. So now our next step is we need to determine the value of D theta by DH when H equals 65. So let's make a note of that. So we're trying to find the value of D theta by DH. When H is equal to 65, so when the height is equal to 65. Fantastic. So, with that said, Now, we need to write, without a doubt, we can go ahead and write that D theta by DH is going to be equal to 90 divided by 90 minus 65 means we're plugging in a value of 65 in for H. So when you take 90 minus 65 squared. Plus 65 squared. So, When we plug this into our calculator, we will find that D theta by DH is going to be equal to when we write our final answer in a fraction form in its most simplified fraction form, you will find that our final answer is equal to 9 divided by 485, and it's units of measurement are going to be in radiance per feet, and that's it. That is our final answer. Hooray, we did it. So, looking back at our problem itself to connect everything that we just They they connect everything that we just did together. Once again, our final answer, without a doubt has to be 9 divided by 485, and don't forget that it's units of measurement are going to be in radiance per beat. So to quickly recap what we've learned, So essentially this problem, in order to solve it effectively and quickly, we need to have a strong understanding of geometry and trig identities to help us know, so to help us solve for this means we need to understand how triangles work, and we can, since we're going to be using triangles to help us solve for our final answer. And we also have to have an understanding of differentiation. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

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Trigonometric Functions

Differentiation

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