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

The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance be- tween the point and the light source. Two lights, one having an in- tensity eight times that of the other, are 6 $\mathrm{m}$ 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 be- tween the point and the light source. Two lights, one having an in-
tensity eight times that of the other, are 6 $\mathrm{m}$ apart. How far from the stronger light is the total illumination least?

Key Concepts

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.

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.

Transcript

00:01 Okay, in this problem, we have two lights.

00:04 And the first light doesn't give off as much light as the second one.

00:10 The second one is eight times stronger.

00:12 So what i'm going to do is i'm going to sign a rating of one to the first one and eight to the second.

00:19 And it tells us the proportion of it is the reciprocal of the square of the distance.

00:27 Okay, so what we're going to do is we're going to assign the distance from the brighter one.

00:32 As x.

00:33 So we're looking for a spot between the two in which the illumination is the least.

00:40 Okay, so let's just pick that spot there.

00:43 And i'm going to sign this a distance of x.

00:47 Well, this distance here would then be six minus x.

00:52 All right.

00:53 And the illumination at this spot between them would be the illumination from both of them added together because you're going to get brightness from both sides.

01:03 So the function i'm going to end up with here is 8 over x squared, okay, the illumination over the distance squared, and then i'm going to add 1 over 6 minus x squared.

01:22 All right.

01:23 So what i want to do is first take the derivative of this function, okay, because i'm looking for a minimum, so i need the derivative.

01:32 So the derivative, we're going to start here with negative 16 x to the negative third.

01:40 What i did was converted this to x to the negative second, took the derivative of that.

01:45 I've negative 16 x to the negative third.

01:48 All right.

01:49 Then i'm going to do the quotient rule here.

01:51 I'm going to have the denominator times the derivative of the numerator.

01:56 Well, the derivative of 1 is just going to be 0, so i'm not going to write that down.

02:01 And then subtract the numerator times the derivative of the denominator...

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