In a rectangle ABCD of width 1 and length 2, suppose AD is a length. Let R be a point selected at random from within the rectangle. The proba- bility that <ARD is obtuse is closest to X/400 where X is an integer. Find X

We are tasked with finding the probability that the angle \( \angle ARD \) is obtuse, where \( A \), \( B \), \( C \), and \( D \) are the vertices of a rectangle with width 1 and length 2. Specifically, the problem asks us to find the integer \( X \) such that the probability is closest to \( \frac{X}{400} \).

### Step 1: Understanding the Geometry
Given the rectangle ABCD, we define the points as follows:
- \( A(0, 0) \)
- \( B(2, 0) \)
- \( C(2, 1) \)
- \( D(0, 1) \)

We want to find the probability that the angle \( \angle ARD \) is obtuse when point \( R(x, y) \) is randomly chosen inside the rectangle. This occurs when the dot product of vectors \( \overrightarrow{RA} \) and \( \overrightarrow{RD} \) is negative:
\[
\overrightarrow{RA} = (-x, -y), \quad \overrightarrow{RD} = (-x, 1 - y)
\]
The dot product is:
\[
\overrightarrow{RA} \cdot \overrightarrow{RD} = (-x)(-x) + (-y)(1 - y) = x^2 + y - y^2
\]
For \( \angle ARD \) to be obtuse, this dot product must be negative:
\[
x^2 + y - y^2 < 0 \quad \Rightarrow \quad y^2 - y - x^2 > 0
\]
This inequality defines the region within the rectangle where \( \angle ARD \) is obtuse.

### Step 2: Finding the Area of the Region
We now need to calculate the area of the region within the rectangle where \( y^2 - y - x^2 > 0 \). This involves integrating the inequality over the bounds of the rectangle:
- \( x \in [0, 2] \)
- \( y \in [0, 1] \)

We solve the inequality \( y^2 - y - x^2 > 0 \) for \( x \). Rearranging, we get:
\[
x^2 < y^2 - y \quad \Rightarrow \quad x < \sqrt{y^2 - y}
\]
This inequality only holds when \( y^2 - y > 0 \), or \( y > 1 \) or \( y < 0 \), which leads

In a rectangle ABCD of width 1 and length 2, suppose AD is a length. Let R be a point selected at random from within the rectangle. The probability that <ARD is obtuse is closest to X/400 where X is an integer. Find X.

Learn how to calculate the probability that the angle ARD in a rectangle is obtuse, using vector dot product and geometric concepts. This solution involves evaluating the region where the dot product is negative and determining the probability. Suitable for advanced high school students (Grades 11-12).

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