6) In a rectangle with an area of $150 \mathrm{~m}^{2}$, the length of the base is equal to $\frac{3}{2}$ of the height. What is the perimeter of the rectangle?
(A) $50 \mathrm{~m}$,
(B) $54 \mathrm{~m}$,
(C) $60 \mathrm{~m}$,
(D) $64 \mathrm{~m}$,
(E) $70 \mathrm{~m}$.
6. The answer is $(\mathbf{A})$.

The rectangle in question can be divided into 6 squares with side length $A H$, where $\overline{A H}=\frac{1}{3} \overline{A B}=\frac{1}{2} \overline{B C}$. The area of the rectangle is therefore $\overline{A B} \cdot \overline{B C}=6 \overline{A H}^{2}=150 \mathrm{~m}^{2}$, which means $\overline{A H}=5 \mathrm{~m}$.

The perimeter of the rectangle is thus $2 \overline{A B}+2 \overline{B C}=6 \overline{A H}+4 \overline{A H}=10 \overline{A H}=50 \mathrm{~m}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_3d973c0a184ce9fc7311g-2.jpg?height=320&width=423&top_left_y=545&top_left_x=1499)

Alternatively, if we denote the length and height of the rectangle by $x$ and $y$ respectively, we have $x y=150 \mathrm{~m}^{2}, x=\frac{3}{2} y$ from which we obtain $y=10 \mathrm{~m}$ and $x=15 \mathrm{~m}$. The perimeter is finally given by $2 x+2 y=50 \mathrm{~m}$.