 Imagine a man is walking along a straight path at a speed of four feet per second. So he's walking along this straight path right here. And so let's make it a little bit more interesting. Let's now make him a prisoner who's breaking out of the clink. Oh no, he's running, running, running, which maybe he should go a little bit faster than four feet per second, but that's beside the point. So a spotlight then catches the man and it's located exactly 20 feet away from the path that he's running along. And by saying that it's 20 feet away from the path, what I mean is it's 20 feet's the shortest distance between the spotlight and the path, which will necessarily be a right angle that you can see here in this diagram. And the spotlight is turning so that the light stays on the man, because you know, if we're gonna release the guns or the dogs or whatever, we need to see this man in the dark because we've now made him into a jailbird. At what rate is the spotlight rotating when the man is 15 feet from the point on the path closest to the searchlight? So you can see the diagram going on right here. So at this moment, the man's running. This distance right here is 15 feet. This distance right here is 20 feet. It does form a right triangle going on here. Some other information we know is the man's running along the path or just walking slowly, I suppose. I mean, four feet per second is not incredible. I mean, it's not horrible either. I mean, it's not crawling, but whatever. It's got the spotlight on him. He might want to change his velocity, but again, this is all not the point here. So we have this velocity moving in the direction there. At what rate is the spotlight rotating? Notice this language here seems to suggest that what we're looking for is a d theta dt. We wanna know at what speed is the angle changing. So this one, because theta is the question we're after, turns out trigonometry is gonna become very important here. If we take x to be the distance between the man's current location and this closest distance to the spotlight right here, if that's x, then what we know is that dx dt is equal to four feet per second. And we're trying to figure out what's the change of angle with respect to time at the moment that x equals 15 feet, like so. So we need to find a connection between x and the angle theta. Well, because we know this distance right here, notice that with respect to theta here, we have the opposite side, we have the adjacent side. The tangent ratio seems to make sense here. So we get that tangent of theta is equal to x over 20. And so with this equation in mind, we then can take the derivative to help us find out the related rate of d theta dt with respect to dx dt. And we're just gonna take the derivative implicitly if we take the derivative of both sides with respect to time, because both the angle and the distance theta and x are changed with respect to time. On the left hand side, when we take the derivative of theta, tangent of theta, we're going to get the derivative of tangent is a secant squared theta, but we have to also take the inner derivative because theta itself is a function of time. And then on the right hand side, when we take the derivative of x over 20, we're gonna get x prime over 20. And so let's then plug in the things we know here. So, well, actually before we do that, let's solve for theta prime. So theta prime is gonna equal x prime over 20. And we're gonna divide this by secant squared theta, which divided by secant, it's the same thing as times it by cosine. So we're gonna get x prime cosine squared theta all over 20. So x prime, we know what that is. That's just equal to a four. So we see that we're gonna get four over 20. But what do we do with this cosine squared? Well, can we calculate cosine theta of this triangle? Notice we don't actually know what theta is, nor do we actually need to know what theta is. We need to know what cosine of theta is. So when you look at this triangle right here, what is cosine of theta? Well, cosine is gonna be adjacent over hypotenuse, which we don't know what the hypotenuse is. Could we figure out the hypotenuse? Absolutely, because after all, we know this side and this side, we can use the Pythagorean equation. Notice that 15 squared plus 20 squared is gonna equal this other side, the hypotenuse squared. In which case, when we work through that, you're gonna get that 15 squared is 225. 20 squared is 400. This gives us the hypotenuse squared. Therefore, the hypotenuse is gonna equal the square root of 225 plus 400, so 625. And the square root of 625 actually turns out just to be 25. Turns out this was just a three, four, five triangle, I just times each side by five, haha. People like to do that all the time just to make the numbers a little bit easier for us because we don't wanna fixate so much on the arithmetic. It's the calculus that matters here. So the hypotenuse of this triangle is 25. The adjacent side is 20 with respect to theta. So cosine is going to be 20 over 25. But given that we're gonna square this puppy in just a moment, it might be easier to simplify the fraction first. 20 over 25 is the same thing as four over five. And so in the end, we end up with a four cubed over 20 times five squared. That becomes a 16 over 125, 125, excuse me, which that would be approximately, throw that in a calculator, you're gonna be approximately 0.128. What are the units in play here? We're talking about a rotating spotlight. The units in play are gonna be radians per second. Measure, the measurement of time here was s, was just seconds, the man was running four feet per second. As we're measuring angles here, the default measurement of angles is gonna be radians. If we wanna convert this into degrees, we can do that, but we have to convert from radiance to degrees. And so we're just gonna leave it at this. In order to keep the spotlight on the man when he's 15 feet away, the spotlight needs to be turning at a rate of 0.128 radians per second.