Task B-2.7. Ivo has decided to build a house on a plot of land that has the shape of a scalene triangle, with a base of $16 \mathrm{~m}$ and a height to that base of $12 \mathrm{~m}$. One wall of the house needs to be built on the base of the triangle. Since the plot is small, he wants to use it as efficiently as possible. The floor plan of the house should be a rectangle of maximum area. Determine the dimensions and area of the floor plan of the house.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_0148fcdd1c92efebb536g-12.jpg?height=767&width=978&top_left_y=652&top_left_x=541)

Sketch . . .

Triangles $\triangle A^{\prime} B^{\prime} C$ and $\triangle A B C$ are similar because all their angles are congruent.

Then,

$$
\frac{\left|A^{\prime} B^{\prime}\right|}{\left|C D^{\prime}\right|}=\frac{|A B|}{|C D|}
$$

i.e.,

$$
\frac{y}{v-x}=\frac{a}{v} \quad \Rightarrow \quad y=\frac{a}{v}(v-x)
$$

It follows that

$$
P(x)=x y=x \frac{a}{v}(v-x)=a x-\frac{a}{v} x^{2}=16 x-\frac{4}{3} x^{2}
$$

This function reaches its maximum at $x=x_{0}=-\frac{16}{2 \cdot \frac{-4}{3}}=6 \text{ m}$. Then,

$$
y=\frac{16}{12}(12-x)=\frac{4}{3}(12-6)=8 \text{ m}
$$

Thus, the width of the house's floor plan is $x=6 \text{ m}$, the length is $y=8 \text{ m}$, and the area is $P=x y=48 \text{ m}^{2}$. (1 point)