Question

The illumination from a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source . Two light sources of intensities I sub 1 and 1 sub 2 are d units apart. What point on the line segment joining to the two sources has the least illumination?

Answer 1

they are asking for the spot between the 2 lamps that has the least light
The total light at a spot between them is

Answer 2

take the derivative and set it = to zero and solve for x

Answer 3

when i already got the x.. is it already done?

Answer 4

if you get the value of x then its over
i think so

Answer 5

Let's use A= \(I_1\) and B= \(I_2\) (easier to type)
take the derivative with respect to x of the equation that gives the total illumination that reaches a spot between the two lamps:
\[ \frac{d}{dx}(\frac{A}{x^2} +\frac{B}{(d-x)^2} )=0\]
use the power rule
\[ dx^n= n x^{n-1} dx\]

\[ \frac{d}{dx} (A x^{-2} + B(d-x)^{-2} )= 0\]
you get
\[ -2 A x^{-3} -2 B (d-x)^{-3}(-1) =0 \]
or
\[ \frac{A}{x^3}= \frac{B}{(d-x)^3} \]
\[ (\frac{(d-x)}{x})^3= \frac{B}{A} \]
take the cube root of both sides
\[ \frac{(d-x)}{x}= \sqrt[3]{\frac{B}{A}}\]
let c= cube root of B/A for convenience:
d-x= cx
x+cx= d
x= d/(1+c)