The height of the top of the ramp is the side opposite the $12^{\circ }$ angle, and the ramp length of 15 feet is the hypotenuse. Use sine: $\sin (12^{\circ })=\frac{\text{opposite}}{15}$. So, $\text{opposite}=15\sin (12^{\circ })\approx 15(0.208)=3.12$. This is about $3.1$ feet, so the correct answer is $3.1$.

Using the definition of tangent, $\tan A=\frac{\text{opposite}}{\text{adjacent}}$. Since $\tan A=\frac{3}{4}$, the ratio of opposite to adjacent sides is $3:4$. The opposite side is given as $12$, so the scale factor is $12÷3=4$. Multiply the adjacent part of the ratio by this scale factor: $4\cdot 4=16$. Therefore, the length of the side adjacent to angle $A$ is $16$.

Since $\tan X=\frac{5}{12}$, the side opposite angle $X$ can be represented as $5$ and the side adjacent to angle $X$ as $12$. In a right triangle, the hypotenuse is found using the Pythagorean theorem: $\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13$. Therefore, $\cos X=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{12}{13}$.

Segment $AB$ is horizontal from $(0,0)$ to $(8,0)$, so its length is $8$. Segment $BC$ is vertical from $(8,0)$ to $(8,6)$, so its length is $6$. This makes a right triangle with right angle at $B$. For $\angle BAC$, the opposite side is $BC=6$ and the adjacent side is $AB=8$. Therefore,

Because $x+y=90$, the angles are complementary, so $y=90-x$. In a right triangle, if $\tan x=\frac{3}{4}$, then the side opposite angle $x$ can be taken as 3 and the adjacent side as 4. Using the Pythagorean theorem, the hypotenuse is $\sqrt{3^2+4^2}=5$. For angle $y$, the side opposite $y$ is the side adjacent to $x$, which has length 4. Therefore, $\sin y=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{4}{5}$. So the correct answer is $\frac{4}{5}$.