The bottom of a large theater screen is 3 ft above your eye le...

Question

The bottom of a large theater screen is 3 ft above your eye level and the top of the screen is 10 ft above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of 3 ft/s while looking at the screen. What is the rate of change of the viewing angle θ when you are 30 ft from the wall on which the screen hangs, assuming the floor is horizontal (see figure)?

Accepted Answer

Below there. Today we're going to 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. Imagine that you find yourself standing in front of a large billboard. The bottom and top edges of the billboard are located 5 and 15 ft above your eye level, respectfully. As you walk directly away from where the billboard is mounted at a speed of 4 ft per second, you continually have the billboard in your line of vision. Determine how quickly the subtended angle theta by the billboard from your eyes is changing when you are located 25 ft away from where the billboard is mounted. Assume the ground you are walking on is flat. Awesome. So it appears for this particular problem were asked to figure out how quickly the subtended angle theta. Bye The billboard from your eyes is changing when you are located 25 ft away from the billboard is originally mounted. So we're trying to figure out Once again, we're trying to figure out this subtended angle theta by the billboard from your eyes is changing. So like a rate of change is what we should be thinking about when you're located 25 ft away from where the billboard is mounted. So that's what we're ultimately trying to solve for. So we're trying to figure out how quickly the subtended angle theta by the billboard from your eyes is changing. Based on all the conditions set to us by this problem itself. Awesome. So with that said, now that we know what we're solving for, let's read off our multiple choice answers to see what our final answer might be, noting that all of our multiple choice answers are in the same units of radiances per second. So A is negative 44 divided by 1,105, B is negative 22 divided by 1,105. C is negative 27 divided by 551, and D is negative 9 divided by 551. OK. So, first off, in order to solve this problem, to make things more easy on ourselves, let's create a diagram to better visualize what is happening in this particular prompt. So I've gone ahead and found a detailed digital or a digitized diagram. So we have We have our billboard, which is represented by this narrow blue rectangle in the far left of our diagram, and as you can see from the bottom and top edges of the billboard are located 5 and 15 ft above your eye level, so we can assume that the height of the person. Maybe about, like, let's say like the average height, like they're 5 ft something. And as we could see this angle theta will be the angle subtended by the billboard at your eye level and let us state that X, as we could see when we labeled it in our diagram, and X is going to be the horizontal distance from where you are standing to the base of the billboard. As we could see, because the base of the billboard when you go down. As you could see, and I'll do a little red dotted line, so it's on the base of the billboard going down to where we're standing. So let's imagine that we are standing. At the bottom, and we're looking up in this angle theta is the angle subtended by the billboard at your eye level. So you're standing and you're looking up at the billboard. Awesome, as you can see, it's triangular in shape, so you could start thinking that we're probably gonna have to use trig identities to help us solve for this particular problem. Awesome. So, using our figure to help us set up our equations, we can recall and note that the angle theta can be expressed as the following using geometry. So we can describe angle theta in the following terms using our trig identities, and we can go ahead and write that theta is going to be equal to inverse tangent of. 15 divided by X. Minus Inverse tangent of 5 divided by X, and we now need to differentiate both sides with respect to time T, so we need to take D theta by DT is equal to D by DT. Of Tangent or inverse tangent of. 15 divided by X. Minus D by DT. Of inverse tangent of 5 divided by X, fantastic. So now we need to recall and apply the chain rule in order to to perform this particular derivation, in order to do the derivative, we need to recall and use chain rule. So we need to recall and note that the derivative of inverse tangent of you with respect to, which we can we can shorten this as WRT with respect to. X will be 1 divided by 1 plus U 2 multiplied by DU by DX and for inverse tangent of, 15 X, we could say that D by DT. Of Inverse tangent of 15 divided by X. is going to be equal to -15. Divided by So negative 15, which you could pull the negative out to the front too, whatever makes the most sense to you. So -15 divided by X2 plus 15 squad multiplied by D by DX. Or rather, more specifically, we could say DX by DT. Fantastic. So, now we need to focus our attention now on. Inverse tangent of drum roll 5 divided by X. So when we do that, we need to take D by DT of Inverse tangent of 5 divided by X, which is equal to negative. 5 divided by X2 + 5 squad multiplied by DX by DT. Fantastic. So, therefore, without a doubt, we can go ahead and write that D theta by DT is going to be equal to negative, and now we could pull out the negative sign to the front, so negative. 15 divided by X2 + 15 squad multiplied by DX by DT. Minus -5 divided by X2 + 52 multiplied by DX by DT. Fantastic. So now we need to substitute in our value for DX by DT and salt. So we need to note that we're told that we're walking away at 4 ft per second, and DX by DT and let's make a note of this. So let's note that DX by DT is going to be equal to 4, because once again we're walking away at 4 ft per second. Fantastic, and we need to note that we're trying to find D theta by DT. When, so this is key, so when. X is equal to 25. Fantastic. So with that said. Our next step now. we need to substitute in the differentiated expressions into our equation, so we need to take D theta by DT and say that it is equal to -15 multiplied by 4 divided by 25 2 + 15 squared. Fantastic. And then we need to. Make sure we subtract -5 multiplied by 4, all divided by 25 2 + 5 squad. So when we plug that big old long expression into our calculator and we leave our answer infraction form, we will find that our final answer has to be equal to -44 divided by 1,105, and don't forget our units of measurement are now in radiance per second. And that is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer, without a doubt has to be multiple choice answer, letter A, which once again states that our final answer is negative 44 divided by 1,105 radiants per second. 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

Implicit Differentiation

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