4.18. A pyramid is intersected by a plane parallel to the base. Find the function expressing the dependence of the area of the section on the distance between the vertex of the pyramid and the intersecting plane.
4.18. By the condition, the height $S O_{1}=h$ (Fig. 4.16) is a variable.

It is known that $\frac{S_{\triangle A_{1} B_{1} C_{1}}}{S_{\triangle A B C}}=\left(\frac{h}{H}\right)^{2}$, i.e., $S_{\triangle A_{1} B_{1} C_{1}}=h^{2} \frac{S_{\triangle A B C}}{H^{2}}$, where $S_{\triangle A B C}$ and $H$ are constant values in this problem.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-580.jpg?height=411&width=411&top_left_y=1158&top_left_x=735)

Fig. 4.16 Thus, the dependence between the area of the section and the distance from the vertex to the cutting plane is expressed by a quadratic function of the form $f(x)=k x^{2}$, where $k=\frac{S_{\triangle A B C}}{H^{2}}>0$ is a parameter, and $x=h$ is the variable; and $0 \leq x<H$.