The intensity of illumination at any point from a light...

Name: Problem 47, Chapter 4: University Calculus: Early Transcendentals
Uploaded: 2020-05-18T19:44:37-08:00
Duration: 8 min 30 s
Channel: Regina Hays
Description: The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an intensity eight times that of the other, are 6 m apart. How far from the stronger light is the total illumination least?

The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an intensity eight times that of the other, are 6 m apart. How far from the stronger light is the total illumination least?

Key Concepts

Superposition Principle

When dealing with multiple sources, the total effect at any point is the sum of the individual effects from each source. This principle is important for combining intensities (or similar effects) from different contributors to determine overall impact.

Calculus-based Optimization

This is the process of using calculus, particularly differentiation, to find the maximum or minimum values of a function. In these types of problems, one formulates an expression for the total effect (such as illumination), differentiates it with respect to a variable, and then sets the derivative to zero to locate critical points that correspond to optimal conditions.

Inverse Square Law

This principle states that a physical quantity (like illumination or gravitational force) is inversely proportional to the square of the distance from the source. It implies that as you move away from the source, the effect diminishes rapidly, which is fundamental in understanding how intensities or fields behave in space.

Proportional Relationships

This concept involves understanding how one quantity changes in relation to another. In this context, it signifies that the illumination is directly linked to a constant factor divided by the square of the distance, allowing the establishment of mathematical models based on direct or inverse proportions.

Transcript

00:02 Okay, we are going to start with two light sources.

00:07 So we have two light sources.

00:14 First one and the second one.

00:19 They are six meters apart.

00:26 And we want to determine that if i pick a point here, somewhere here let this be x and then this be x and this is 6 minus x and we have a point right here somewhere in between the two light sources how far from the stronger light is the illumination the least so we want the minimum illumination from strongest light source.

01:41 And so we're going to say that this one has is eight times the intensity as the other one.

01:58 Okay.

01:59 And what we do know is that intensity is actually proportional.

02:08 So we have a proportionality constant.

02:14 The square, the square of the reciprocal of the distance between the point and the light source.

02:25 So the square of the light source.

02:31 Okay.

02:32 So we're going to put that in terms of both of these light sources.

02:38 So intensity is actually equal to the, or is proportional to the square of the reciprocal of the distance.

02:47 Okay.

02:48 So if i actually take it in terms of the situation, we know that the intensity of the first light source to the point is that proportionality times 6 minus x squared.

03:08 Added to, because not only is this point being illuminated by the first light source, he's also being illuminated by the second light source.

03:19 And the second light source has eight times that intensity, so that is going to be 8k over x squared.

03:30 So that's what we have.

03:32 Okay...

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