Let A and B be fixed points in the plane with distance AB = 1. An ant walks on a straight
line from point A to some point C in the plane and notices that the distance from itself to B
always decreases at any time during this walk. Compute the area of the region in the plane
containing all points where point C could possibly be located.
1. **Understanding the Problem:**
   - We are given two fixed points \( A \) and \( B \) in the plane with \( AB = 1 \).
   - An ant walks from \( A \) to some point \( C \) such that the distance from \( C \) to \( B \) always decreases.
   - We need to find the area of the region in the plane where point \( C \) could be located.

2. **Geometric Interpretation:**
   - The condition that the distance from \( C \) to \( B \) always decreases implies that the ant must be moving directly away from \( B \).
   - This means that the line segment \( AC \) must be tangent to the circle centered at \( B \) with radius \( AB = 1 \).

3. **Circle and Tangent Line:**
   - Draw a circle centered at \( B \) with radius \( 1 \).
   - The point \( C \) must lie on the line that is tangent to this circle at point \( A \).

4. **Angle Consideration:**
   - The tangent line at point \( A \) makes a \( 90^\circ \) angle with the radius \( AB \).
   - Therefore, the region where \( C \) can be located is the half-plane on the side of the tangent line that does not include \( B \).

5. **Area Calculation:**
   - The region where \( C \) can be located is a half-plane bounded by the tangent line at \( A \).
   - However, since the distance from \( C \) to \( B \) must always decrease, \( C \) must lie within the quarter circle centered at \( A \) with radius \( 1 \) and extending in the direction away from \( B \).

6. **Final Area:**
   - The area of a quarter circle with radius \( 1 \) is given by:
     \[
     \text{Area} = \frac{1}{4} \pi r^2 = \frac{1}{4} \pi (1)^2 = \frac{\pi}{4}
     \]

The final answer is \(\boxed{\frac{\pi}{4}}\).