To solve the problem, we need to find the probability that the angle $\angle ARD$ is obtuse for a randomly chosen point $R$ inside the rectangle $ABCD$.

Step 1: Analyze the Geometry

Given:

• Rectangle $ABCD$ has width $1$ and length $2$, so $AB = 2$ and $AD = 1$.
• Point $R$ is selected randomly within the rectangle.

We want to find the probability that $\angle ARD$ is obtuse. An angle is obtuse if the dot product of the vectors forming that angle is negative.

Step 2: Consider the Vectors and Conditions

Let's place the rectangle in the coordinate plane with $A$ at the origin:

• $A = (0, 0)$
• $B = (2, 0)$
• $C = (2, 1)$
• $D = (0, 1)$

For a point $R = (x, y)$ inside the rectangle, the vectors $\overrightarrow{AR}$ and $\overrightarrow{DR}$ are:

• $\overrightarrow{AR} = (x, y)$
• $\overrightarrow{DR} = (x, y-1)$

The angle $\angle ARD$ is obtuse if $\overrightarrow{AR} \cdot \overrightarrow{DR} < 0$, i.e., the dot product is negative.

The dot product is: $\overrightarrow{AR} \cdot \overrightarrow{DR} = x \cdot x + y \cdot (y - 1) = x^2 + y^2 - y$ We need to find where this expression is negative: $x^2 + y^2 - y < 0$ $x^2 + (y - \frac{1}{2})^2 < \frac{1}{4}$ This is the equation of a circle with radius $\frac{1}{2}$ and center $(0, \frac{1}{2})$.

Step 3: Calculate the Area Where $\angle ARD$ is Obtuse

The circle lies within the rectangle, and we need to calculate the area of the circle that is inside the rectangle.

• The area of the full circle is $\pi \times \left(\frac{1}{2}\right)^2 = \frac{\pi}{4}$.
• Since the center of the circle lies on the edge of the rectangle, exactly half of the circle lies within the rectangle.

Thus, the area of the circle within the rectangle is $\frac{\pi}{8}$.

Step 4: Calculate the Probability

The area of the rectangle $ABCD$ is $2 \times 1 = 2$.

The probability that $\angle ARD$ is obtuse is the ratio of the area of the circle to the area of the rectangle: $\text{Probability} = \frac{\frac{\pi}{8}}{2} = \frac{\pi}{16}$ We are given that this probability is $\frac{X}{400}$, so: $\frac{\pi}{16} = \frac{X}{400}$ Solving for $X$: $X = \frac{400 \times \pi}{16} = 25\pi$ Using $\pi \approx 3.14159$, we get: $X \approx 25 \times 3.14159 = 78.54 \approx 79$

Final Answer

So, $X$ is closest to $\boxed{79}$.

Do you want further details, or do you have any questions?

Here are some related questions:

1. How do you determine the area of a circle within a different polygon?
2. What is the importance of the dot product in determining angles?
3. How does the probability change if the shape of the region changes?
4. How would the solution differ if $R$ were chosen from another shape?
5. What if the point $R$ had to be at a specific distance from the vertex?

Tip: For complex problems, breaking down into geometric analysis and vector algebra often simplifies the solution.