To determine the shape of the area (bubble) where the potential field [tex]\( h(p) > 100 \)[/tex], let's break down the given problem step by step:
1. Potential Field Representation:
The potential field [tex]\( h(p) \)[/tex] generated by a kernel located at [tex]\( c \)[/tex] is defined by the function:
[tex]\[
h(p) = 200 - |p - c|
\][/tex]
where [tex]\(|p - c|\)[/tex] denotes the distance between the position [tex]\( p \)[/tex] and the kernel [tex]\( c \)[/tex].
2. Condition for the Potential Field:
We need to find the region where [tex]\( h(p) > 100 \)[/tex].
3. Setting Up the Inequality:
Substituting [tex]\( h(p) \)[/tex] into the inequality, we get:
[tex]\[
200 - |p - c| > 100
\][/tex]
4. Solving the Inequality:
Rearrange the inequality to isolate the distance term:
[tex]\[
200 - 100 > |p - c|
\][/tex]
[tex]\[
100 > |p - c|
\][/tex]
This simplifies to:
[tex]\[
|p - c| < 100
\][/tex]
5. Interpreting the Inequality:
The inequality [tex]\( |p - c| < 100 \)[/tex] represents a circular area centered at the kernel [tex]\( c \)[/tex] with a radius of 100 units. This defines the bubble's shape.
6. Locations of the Kernels:
- The first kernel is located at [tex]\( c_1 = (200, 200) \)[/tex].
- The second kernel is located at [tex]\( c_2 = (200, 400) \)[/tex].
7. Identifying the Bubbles:
- For the kernel at [tex]\( c_1 \)[/tex]:
The bubble is a circle centered at [tex]\( (200, 200) \)[/tex] with a radius of 100 units.
- For the kernel at [tex]\( c_2 \)[/tex]:
The bubble is a circle centered at [tex]\( (200, 400) \)[/tex] with a radius of 100 units.
8. Summary of Bubble Descriptions:
Therefore, the shapes of the areas (bubbles) that satisfy [tex]\( h > 100 \)[/tex] are:
- A circle with center at [tex]\( (200, 200) \)[/tex] and radius 100.
- A circle with center at [tex]\( (200, 400) \)[/tex] and radius 100.
The shapes of these bubbles can be summarized as follows:
[tex]\[
(\{ \text{center}: (200, 200), \text{radius}: 100 \}, \{ \text{center}: (200, 400), \text{radius}: 100 \})
\][/tex]
So, we conclude that the regions where [tex]\( h > 100 \)[/tex] are two circular areas, each with a center at one of the kernel locations and a radius of 100 units.
