This problem. We're looking at a bird flying from an island across the river to its nest. So let's begin with a picture. So we have an island and we'll call that island A. It is directly across the water from B, and it wants to make it all the way to its nest at D. So it's flight pattern is going to be directly across the river at some angle and then across the land over to D, and we'll call this point. See, just we have something we can call it Now we know that B and D have a has a distance of 13 factor in kilometers, so 13 kilometers. So let's call BC Let's call this X Well, if that's X, then from C to D is 13 minus X, and we know that the island is five kilometers from the shore. So we have just about all of our things marked here. Let's, uh let's also just use, um we'll call this why that distance from eight. To see that he's going to fly now what we're wanting to minimize here, we're told that the birds instinctively are going to minimize their energy that they need to expend. So let's look at the energy expended in this situation. Well, the energy expended across the water is going to be, um, energy Times distance. Now we're told that the energy to get across the water is 1.4 times more than than on the land. So if I call the energy across the land one, then the energy for why is going to be 1.4 units and however we're measuring that So the energy across the water is going to be 1.4 times why our energy across the land is going to be 13 minus x times the distance, our times, the energy expenditure on the land. Well, we said that was just gonna be one unit. So across the water is 1.4 across the land is gonna be one. So here is my energy expenditure formula. But the problem is, it's in two variables. I have why, and I have an ex. So let's put them in terms of each other. Take a look at my triangle. I can use the Pythagorean theorem to say that why squared equals five squared plus x squared, or why equals the square root of 25 plus X squared so I can plug that into my equation. So I have 1.4 times 25 plus X squared plus 13 minus x. That is my energy expenditure given in one variable instead of two. If I want to minimize this these air optimization problems, I'm going to take a derivative, set it equal to zero, and that will give me my critical points that show me where I can minimize my formula. My function here. So if I take the derivative that's going to give me well, I have a radical so that 1.4 is on the top. My denominator is going to be two times my radical and in the numerator, I'm gonna multiply by the derivative of what's under the radical sign, which is going to be two x derivative 13 is zero derivative of negative X is negative one. Now, I can do a little bit of simplification here. Not much, but a little bit. And in order to, um, find my minimum energy, I'm going to set this equal to zero. So what I get is 1.4 x over the square root of 25 plus X squared, and that equals one. Hey, now to solve this for X, let's multiply both sides by that radical that's gonna give me 1.4 X equals the square root of 25 plus X squared, comparing a little bit more room to get rid of the radical. We're going to square both sides. So the square of 1.4 is 1.96 x squared and squaring the radical gives me what's underneath Now it's just a matter of algebra combined, my like terms. So I'm going to move my X squared term and divide by 0.96 Now, when I take the square root algebraic Lee, I'm going to get a plus or a minus. But if you look at the problem X is a positive number, The smallest X could possibly be a zero, and that would be straight across the water. X is not gonna have a negative value here, so I could take the positive square root, which is going to be approximately 5.1 kilometers. Okay, so I'm gonna the bird is going to fly across the water, See, is going to be at a 0.5 point one kilometers from B and then continue the rest of the way on two D that will minimize the birds energy expenditure. Now let's tweak this problem a little bit. Let's make it more generic. Instead of saying that it's 1.4 times more energy on water than on land, let's just use I'll use blue here. Let's use W for the energy expenditure on the water an l for the expenditure on the land. So let's revamp this problem again. But using these values. So what is that going to change? Well, I am gonna scroll down in a moment and rewrite, but for the moment, let's look at what I have here. The difference in my original equation instead of 1.4. This is going to be W. That's That's, um that's my expenditure on water, and instead of one, I'm going to have an l. Everything else is going to be the same. So let me scroll down and rewrite this. I'm gonna have my energy expenditure is w times Why, that's my expenditure. Energy expenditure over the water plus 13 minus x times l. That's my energy expenditure overland. So just like before. I'm going to put this in terms of one variable. So I'm gonna have w times the square root of 25 plus x squared That peace has not changed in this problem. And I have 13 minus x times L So let's take the derivative. Well, remember, W and l are constant, so they're not going to change that. My variable is only X. So just like before, taking the derivative of a radical and the denominator is going to be two times the radical W will be in the numerator and we're going to multiply that by the derivative of what's underneath the radical sign. Okay, here l is my constant. So it's going to be l times the derivative of what's in the parentheses, which is negative. One. No, my choose cancel. And what I have is a derivative of E is W X over the square root of 25 plus X squared minus l. Then let's set that equal to zero. That's going to give me W X over my radical equals. L and I would like to find w over l. So let's do a little bit of algebra here. Cross multiply. And if I want w over l that's going to equal 25 plus X squared over X. Now, we're going to use this a little bit, but let's just talk a bit about what this means. How are w and L related? Well, the bigger W is never w is our energy expenditure over water, the more taxing it is to fly over water versus land, the less time the bird will spend flying over the water. So the bigger W is a bigger this ratio is X is gonna become smaller and smaller. We wanna have our Yes, we don't have as much time on land as possible. So a big W means that X is going to get very, very small my way to get off the water as fast as I can. The smaller this ratio is if the energy expenditure over water isn't that much bigger compared toa land, then why not take a more direct route? The smaller that ratio is, the longer my water travel will be because that's the more direct route. So X is gonna be bigger. So that's how w NL and ex are all related to each other. So let's do a couple of examples using this information first. What will the value of W over L. B if the bird goes directly to its nesting area? Well, if it goes directly to its nesting area, that means X equals that entire 13 kilometer unit that means X equals 13. So what is W over l If X is 13? Well, that gives me 25 plus 13 squared, which is 169 over 13. Plug that into my calculator and I get approximately 1.7 is my ratio. Well, what happens if my bird flies directly across the water and then across the land? That means if you look at my picture, if I go directly across the layoff across the water X is going to be zero, there is no triangle. So what happens if I let X equal zero? This was X equals 13. What effect? Zero. Well, that means I'm going toe have the square root of 25 over zero. You can't divide by zero. There is no scenario in which this ratio gets so out of whack that the best thing for the bird to do is to fly directly across the water. It's always going to go at some angle, even for a very large ratio. So that case is not gonna happen. But it is possible that it could go directly to its nesting area. Okay, last question. Ornithologists Observe that certain birds go to shore at a 0.4 kilometers from B. So other words they have found there's a certain bird that when they do this trip, X equals four. So what does that tell me about this ratio of w tell my nexus four. That tells me I have a square root of 25 plus 16/4, which is approximately 1.6. So that is the ratio of my energy expenditure on water versus land for that particular bird.