Exercise 144Calculus › Single-variable calculus › Differentiation › Optimisation and curve sketchingL2Added 4 Nov 2026

At which height should a lamp be hung?

#optimisation#physics#illumination#Lambert's law

Statement

A lamp hangs at height $h$ above a table. At a point of the table at horizontal distance $a$ from the foot of the lamp, the illumination is $I=\dfrac{k\cos\theta}{d^2}$ , where $d$ is the distance to the lamp and $\theta$ the angle between the light ray and the vertical. At which height is the illumination at this point maximal?

Hint

Express $d$ and $\cos\theta$ using $a$ and $h$ : $d^2=a^2+h^2$ and $\cos\theta=\frac hd$ .

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Solution

Illumination as a function of $h$ .
$I(h)=\frac{k\,h}{(a^2+h^2)^{3/2}},\qquad h>0.$

Derivative.
$I'(h)=k\,\frac{(a^2+h^2)-3h^2}{(a^2+h^2)^{5/2}}=k\,\frac{a^2-2h^2}{(a^2+h^2)^{5/2}}.$
Positive for $h<\frac a{\sqrt2}$ and negative afterwards.

Answer. The best height is $h=\dfrac a{\sqrt2}\approx0.71\,a$ , and
$I_{\max}=\frac{2k}{3\sqrt3\,a^2}.$

Remark. Two effects compete: a high lamp is far (the $\frac1{d^2}$ factor), a low lamp shines at a grazing angle (the $\cos\theta$ factor). The compromise is not at all obvious without calculus.

At which height should a lamp be hung?

Statement

Solution

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