Example 3. Using a car crane with a height of 1.5 meters and a boom length of 15 meters, to lift a cylindrical oil tank with a diameter of 6 meters and a height of 2 meters. To find the maximum height at which the car crane can lift the oil tank, it is necessary to first determine the functional relationship between the lifting height of the oil tank and the boom angle $\phi$. Try to find it.
Solve as shown in the figure, according to the algebraic sum of line segments:
$$
\begin{array}{l}
\mathrm{h}=\mathrm{AB}+1.5 \text {. } \\
\text { but } \mathrm{AB}=\mathrm{AD} \\
-\mathrm{BC}-\mathrm{CD},
\end{array}
$$

and $\mathrm{AD}=\mathrm{ED} \sin \varphi$
$$
=15 \sin \varphi \text {, }
$$
$$
\begin{aligned}
\mathrm{BC}=2, & \mathrm{CD}=\mathrm{CF} \operatorname{tg} \varphi=3 \operatorname{tg} \varphi, \\
\therefore \quad \mathrm{AB} & =15 \sin \varphi-3 \operatorname{tg} \varphi-2,
\end{aligned}
$$

thus $\mathrm{h}=15 \sin \varphi-3 \operatorname{tg} \varphi-\frac{1}{2}$, the domain is $0 \leqslant \varphi<\frac{\pi}{2}$.