A rock is whirled horizontally in a circular counterclockwise path about the origin. When the string breaks, the rock travels on a linear path perpendicular to the radius OP and hits a wall located at y = x + 12 If the string breaks when the rock is at P(4,3), determine the point where the rock hits the wall.

To determine where the rock hits the wall after the string breaks, we follow these steps:

### Step 1: Determine the direction of the rock's velocity when the string breaks.
Since the rock is moving in a circular counterclockwise path, its velocity at any point is tangent to the circle at that point. When the string breaks at point \( P(4, 3) \), the velocity of the rock is perpendicular to the radius \( OP \). 

The vector \( \overrightarrow{OP} \) can be calculated as:
\[
\overrightarrow{OP} = \langle 4, 3 \rangle
\]

The velocity vector \( \overrightarrow{v} \) is perpendicular to \( \overrightarrow{OP} \), so we can find \( \overrightarrow{v} \) by swapping the components of \( \overrightarrow{OP} \) and changing the sign of one component. 

This gives:
\[
\overrightarrow{v} = \langle -3, 4 \rangle
\]
This means the rock's path after the string breaks follows the line with a slope given by:
\[
\text{slope of path} = \frac{4}{-3} = -\frac{4}{3}
\]

### Step 2: Write the equation of the line representing the rock's path.
The point-slope form of the line equation is:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope, and \( (x_1, y_1) \) is the point where the string breaks.

Substituting \( m = -\frac{4}{3} \) and \( P(4, 3) \), we get:
\[
y - 3 = -\frac{4}{3}(x - 4)
\]

Expanding this:
\[
y - 3 = -\frac{4}{3}x + \frac{16}{3}
\]

Adding 3 to both sides:
\[
y = -\frac{4}{3}x + \frac{16}{3} + 3
\]

Converting 3 to a fraction with a denominator of 3:
\[
y = -\frac{4}{3}x + \frac{16}{3} + \frac{9}{3}
\]

Simplifying:
\[
y = -\frac{4}{3}x + \frac{25}{3}
\]

### Step 3: Find the intersection of the rock's path with the wall.
The wall is described by the equation \( y = x + 12 \).

To find the intersection, set the equations equal to each other:
\[
-\frac{4}{3}x + \frac{25}{3} = x + 12
\]

Multiply the entire equation by 3 to eliminate the fractions:
\[
-4x + 25 = 3x + 36
\]

Add \( 4x \) to both sides:
\[
25 = 7x + 36
\]

Subtract 36 from both sides:
\[
-11 = 7x
\]

Solve for \( x \):
\[
x = -\frac{11}{7}
\]

Substitute \( x = -\frac{11}{7} \) back into the wall's equation \( y = x + 12 \):
\[
y = -\frac{11}{7} + 12 = -\frac{11}{7} + \frac{84}{7} = \frac{73}{7}
\]

### Step 4: Conclusion
The rock hits the wall at the point \( \left(-\frac{11}{7}, \frac{73}{7}\right) \).

Would you like further details or have any questions?

Here are some related questions:
1. How is the direction of the velocity vector determined in circular motion?
2. What is the significance of using the point-slope form in linear equations?
3. How does the concept of vectors apply to this problem?
4. What are other possible scenarios for the rock's motion if the string breaks at different points?
5. Can you explain why the rock's velocity is tangent to the circle?

**Tip:** When dealing with problems in circular motion, always remember that the velocity vector is perpendicular to the radius at any point on the circle.

Explore how to solve a circular motion problem where a rock breaks free and travels along a linear path perpendicular to its radius vector OP at point P(4, 3). Understand the concepts of circular motion, vector analysis, and linear equation solutions in this advanced high school-level problem.

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