A laser pointer is placed on a platform that rotates at a...

A laser pointer is placed on a platform that rotates at a rate of 20 revolutions per minute. The beam hits a wall $8 \mathrm{~m}$ away, producing a dot of light that moves horizontally along the wall. Let $\theta$ be the angle between the beam and the line through the searchlight perpendicular to the wall (Figure 11). How fast is this dot moving when $\theta=\frac{\pi}{6}$ ?

Key Concepts

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to time based on the rate of change of another interdependent quantity. Such problems typically require setting up an equation that links the two variables and then differentiating with respect to time to relate the rates of change.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. In related rates problems, it is often necessary to differentiate an intermediate variable with respect to another variable and then multiply by the derivative of that variable with respect to time to obtain the overall rate of change.

Trigonometric Differentiation

Trigonometric differentiation refers to finding derivatives of trigonometric functions, such as the tangent or secant functions. In many geometric problems, these functions are used to relate angles to distances, and their derivatives are key in establishing the relationship between angular and linear rates of change.

Angular and Linear Motion

Angular and linear motion are connected via geometric relationships, where an object's rotational movement is converted into linear motion based on the distance from the axis of rotation. Understanding how angular velocity translates to linear speed is essential in problems where rotations lead to movement along a straight line, such as a laser dot moving along a wall.

Transcript

00:01 For this problem, i have a laser which is rotating and putting a reflection up on a wall.

00:07 So let's take a moment and draw our picture.

00:10 Even if you're not a great artist, it's important to draw pictures with related rate problems.

00:16 So you can see how all of your numbers and letters relate to each other.

00:21 Okay, so here's my picture.

00:22 My wall's at the top.

00:24 My laser is a fixed point, but it is spinning.

00:28 Okay, around.

00:29 And so the laser is rotating in circles.

00:33 On my picture, if i have a fixed number that's a constant, i'm going to put it on, and i'm going to label it.

00:40 If a number is changing, i'm going to refer to it with a variable.

00:43 I don't want any changing numbers to show up on my diagram here.

00:49 So what do i have? well, i do have one fixed number.

00:52 The laser is positioned eight meters from the wall.

00:56 What is changing is the angle because it is rotating.

01:01 And the position of the laser dot on the wall as the laser spins, x is constantly moving.

01:08 My laser point on the wall is moving.

01:13 What do i know? i know that my laser pointer is rotating at a rate of 20 revolutions per minute.

01:21 So the change of fade, the change of my angle, is 20 revolutions per minute.

01:29 Now, when i talk about an angle, i don't really want this to be in terms of revolution.

01:33 I would like it to be in terms of radiance.

01:36 So let's do a quick unit change.

01:39 I want to get rid of revolutions and put it in terms of radians instead.

01:44 One revolution is 2 pi radiance.

01:49 So my revolutions cancel.

01:52 So this is 40 pi radians permanent.

01:57 Okay, that's d theta dt.

01:59 That's how fast that angle is changing.

02:01 And i also know that i'm looking at the specific point.

02:05 When i get there, the specific point when theta equals pi over six.

02:10 So i will be looking at that particular point.

02:13 Now, what's my actual question? the question is how fat is the dot moving, the laser dot on the wall? okay.

02:21 So how fast is it moving? that's how fast this position is changing...

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