Optimizing Illumination: Finding the Minimum Point Between Two Light Sources

[Feodalherren]

Homework Statement

The illumination of an object is directly proportional to to the strength of the source and inversely proportional to the distance squared. If two light sources, one three times stronger than the other, are placed 10 ft apart, where should and object be places on the line between the sources so as to receive the least illumination?

Homework Equations

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The Attempt at a Solution

I = s / d^2

[S/d^2] + [3S/(10-d) ^2] = I <-- minimize

I'm completely stuck here. Am I treating d as a constant and just doing d/dS or what?

Sorry that's confusing. I should call distance X instead. Am I taking the derivative with respect to distance was what I meant to ask.

 

[LCKurtz]

Homework Statement

The illumination of an object is directly proportional to to the strength of the source and inversely proportional to the distance squared. If two light sources, one three times stronger than the other, are placed 10 ft apart, where should and object be places on the line between the sources so as to receive the least illumination?

Homework Equations

-----

The Attempt at a Solution

I = s / d^2

[S/d^2] + [3S/(10-d) ^2] = I <-- minimize

I'm completely stuck here. Am I treating d as a constant and just doing d/dS or what?

Sorry that's confusing. I should call distance X instead. Am I taking the derivative with respect to distance was what I meant to ask.

Yes, it would be better to call the distance from one source $x$ and the other $10-x$ to not confuse it with the $d$ in derivatives. So you have $I$ as a function of $x$. You have left out the constant of proportionality, which shouldn't matter anyway. So use calculus to find what value of $x$ gives the minimum value of $I(x)$.