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# Two light sources of identical strength are placed $10$ m apart. An object is to be placed at a point $P$ on a line $\ell$, parallel to the line joining the light sources and at a distance $d$ meters from it (see the figure). We want to locate $P$ on $\ell$ so that the intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source.(a) Find an expression for the intensity $I(x)$ at the point $P$.(b) If $d = 5$ m, use graphs of $I(x)$ and $I'(x)$ to show that the intensity is minimized when $x = 5$ m, that is, when $P$ is at the midpoint of $\ell$.(c) If $d = 10$ m, show that the intensity (perhaps surprisingly) is not minimized at the midpoint.(d) Somewhere between $d = 5$ m and $d = 10$ m there is a transitional value of $d$ at which the point of minimal illumination abruptly changes. Estimate this value of $d$ by graphical methods. Then find the exact value of $d$.

## a) $I(x)=\frac{k}{x^{2}+d^{2}}+\frac{k}{(10-x)^{2}+d^{2}}=\frac{k}{x^{2}+d^{2}}+\frac{k}{x^{2}-20 x+100+d^{2}}$b) $I(x)$ has a minimum at $x=5 \mathrm{m}$c) $I(x)$ has is minimized at $x=0$ and $x=10 \mathrm{m} .$ And the midpoint is the most brightly lit point.d) 5$\sqrt{2}$ which is approximately 7.07

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