There is a wooden block where $ACFD$ and $ABED$ are two congruent isosceles trapezoids, and $BCFE$ is a rectangle. Let the projection of point $A$ on line $BC$ be $M$ and its projection on plane $BCFE$ be $P$. Given that $\overline{AD} = 30$, $\overline{CF} = 40$, $\overline{AP} = 15$, and $\overline{BC} = 10$. Place plane $BCFE$ on a horizontal table, and call any plane parallel to $BCFE$ a horizontal plane. Let $Q$ be a point on $\overline{FC}$ such that $\overrightarrow{AQ}$ is parallel to $\overrightarrow{DF}$. Using the fact that $\triangle ABC$ and $\triangle ACQ$ are congruent triangles, prove that if a horizontal plane $W$ lies between $A$ and $P$ and is at distance $x$ from $A$, then the rectangular region formed by the intersection of $W$ with this wooden block has area $20x + \frac{4}{9}x^2$. (Non-multiple choice question, 4 points)
There is a wooden block where $ACFD$ and $ABED$ are two congruent isosceles trapezoids, and $BCFE$ is a rectangle. Let the projection of point $A$ on line $BC$ be $M$ and its projection on plane $BCFE$ be $P$. Given that $\overline{AD} = 30$, $\overline{CF} = 40$, $\overline{AP} = 15$, and $\overline{BC} = 10$. Place plane $BCFE$ on a horizontal table, and call any plane parallel to $BCFE$ a horizontal plane.\\
Let $Q$ be a point on $\overline{FC}$ such that $\overrightarrow{AQ}$ is parallel to $\overrightarrow{DF}$. Using the fact that $\triangle ABC$ and $\triangle ACQ$ are congruent triangles, prove that if a horizontal plane $W$ lies between $A$ and $P$ and is at distance $x$ from $A$, then the rectangular region formed by the intersection of $W$ with this wooden block has area $20x + \frac{4}{9}x^2$. (Non-multiple choice question, 4 points)