Irem has a cylindrical container completely filled with water and a marble block in the shape of a cylinder with height equal to half the height of this container. When the marble block is placed in the container so that it is completely submerged in water, $\frac{1}{32}$ of the water in the container overflows.
Accordingly, what is the ratio of the base radius of the container to the base radius of the marble block?
A) 2
B) 3
C) 4
D) $2\sqrt{2}$
E) $4\sqrt{2}$

Let $x$ and $y$ be positive real numbers such that
$$\begin{aligned} & x ^ { 2 } + 3 y ^ { 2 } = 8 \\ & 2 x ^ { 2 } + y ^ { 2 } = 6 \end{aligned}$$
What is the product $x \cdot y$?
A) 2
B) 4
C) 6
D) 8
E) 10

Let $x$ and $y$ be real numbers,
$$\begin{aligned} & x^{2} + 8xy = 60 \\ & y^{2} - 3xy = -15 \end{aligned}$$
Given that this holds, what is the product $\mathrm{x} \cdot \mathrm{y}$?
A) 3 B) 4 C) 5 D) 6 E) 7

In a rectangular coordinate plane, what is the area of the triangular region bounded by the lines $2x - y = 0$, $x + 2y = 0$ and $x - 8y + 30 = 0$ in square units?
A) 9 B) 12 C) 15 D) 18 E) 21