## Maximum Height of a Tilting Refrigerator

Suppose a refrigerator of height
$h$
and depth
$d$
is being tilted. As it tilts its apparent height increases. What angle of tilt will result in the greatest apparent height?

We can consider the apparent height of the refrigerator as being made up of two parts, one due the base and the other due to the vertical height.

The apparent height is
$hcos \alpha + dsin \alpha$
.
Let
$f(\alpha )=hcos \alpha + dsin \alpha$
.
Then
$\frac{df}{d \alpha}=-hsin \alpha + dcos \alpha =0 \rightarrow tan \alpha = \frac{d}{h}$
.
$\frac{d^2f}{d \alpha^2}=-hcos \alpha - dsin \alpha \lt 0$
since
$\alpha$
is acute. The height is a maximum and
\begin{aligned} f(\alpha)_{MAX}=d sin \alpha_{MAX}+hcos \alpha_{MAX} &= d/(cosec \alpha_{MAX})+h/(sec \alpha_{MAX}) \\ &= d/ \sqrt{1+cot^2 \alpha_{MAX}}+h/ \sqrt{1+tan \alpha_{MAX}} \\ &=d / \sqrt{1+h^2/d^2}_h/ \sqrt{1+d^2/h^2} \\ &= d^2/ \sqrt{d^2+h^2}+h^2/ \sqrt{d^2+h^2} \\ &= \sqrt{h^2+d^2} \end{aligned}