In Figure 23.12, a coin is right up against the far edge of the mug. In picture (a) the coin is

In Figure 23.12, a coin is right up against the far edge of...

Key Concepts

Geometric Tangency and Visibility

The concept of tangency is important for determining the exact moment when an object starts to become visible or hidden. In this context, a line of sight that is tangent to an object signifies a critical point where the object is just obscured or just revealed. Understanding this helps in setting up the correct geometric relations needed to solve for the dimension of the object in question.

Perspective Projection

This concept explains how the size, shape, and position of objects appear altered to an observer depending on their distance from the viewer. In problems like these, understanding perspective is essential as it governs how much of an object is visible when it is partially obscured by another object, and why objects farther away or behind an obstruction appear smaller or hidden.

Similar Triangles

Similar triangles are a fundamental geometric tool used to relate the dimensions of various parts of a diagram through proportional relationships. When lines of sight, object edges, and boundaries intersect, the triangles formed are similar, meaning that corresponding side ratios are equal. This property is crucial for calculating unknown lengths or diameters based on known distances and dimensions.

Transcript

00:01 Okay, i've got a rough sketch of this situation here, and i'll explain my sketch.

00:05 We've got the mug that is 8 .9 centimeters tall.

00:09 And when that mug has nothing in it, the coin is out of view.

00:16 And we use that information to get this straight line from the eye down here to point b for the width of the mug, which is 6 .5 centimeters.

00:24 And when that mug is filled with water, the rays or the, um, the, the, um, the, the, the, um, the, the view that we have is refracted due to the index of refraction with water and how it bends the rays.

00:38 So we're going to use that information in a little bit of geometry in order to figure out the diameter of this coin that comes into view when the mug is filled with water.

00:48 So to get started, we want to figure out this length right here of the hypotenuse in order to use and then we use that information in order to figure out this angle of reflection here.

01:05 This angle of reflection is equal to this angle down here.

01:10 They're just complementary angles or congruent angles.

01:15 So we'll use pythagorean's theorem to solve for the length of a, b.

01:22 Pythagorean's theorem is one side squared plus the other side squared.

01:28 Is equal to the hypotenuse squared.

01:33 And we can rearrange this to solve for the length of the hypotenuse square root of a .c squared plus cb squared.

01:45 And then when i substitute in my values, 8 .9 squared plus 6 .5 squared.

01:55 And you square root all of that, you get 11 .01 centimeters.

02:00 So that is the length.

02:02 Of our hypotenuse.

02:04 Now we can use that information.

02:06 I want you to recall something from way back when called sokotoa.