Find the arc length from 

(0, 7)

 clockwise to 

(6, 

13

)

 along the circle 

*x*2 + *y*2 = 49.

To find the arc length along the circle from point \( (0, 7) \) to \( (6, 13) \) in the clockwise direction, let's break the problem down step by step:

### Step 1: Equation of the Circle
The given equation of the circle is:
\[
x^2 + y^2 = 49
\]
This equation represents a circle centered at \( (0, 0) \) with a radius of \( r = \sqrt{49} = 7 \).

### Step 2: Convert the Points to Polar Coordinates
The arc length between two points on a circle can be found using their angular positions in polar coordinates.

- The point \( (0, 7) \) is directly on the positive \( y \)-axis, which corresponds to an angle of \( \theta_1 = \frac{\pi}{2} \) radians (90 degrees).
- For the point \( (6, 13) \), we first need to verify that it lies on the circle, which we can check by substituting its coordinates into the circle equation:
\[
6^2 + 13^2 = 36 + 169 = 205 \neq 49
\]
So, the point \( (6, 13) \) does not lie on the given circle, which means we cannot proceed with the arc length calculation using this point.

Please verify if there's any mistake with the second point, or would you like to modify the coordinates to match a point on the circle? Let me know if you'd like to proceed with a different set of coordinates!

Find the arc length from (0, 7) clockwise to (6, 13) along the circle x^2 + y^2 = 49.

Explore how to calculate the arc length on a circle from point (0, 7) to (6, 13) using polar coordinates and the circle equation x^2 + y^2 = 49. This problem highlights key concepts in geometry and polar coordinate systems. Suitable for high school students in Grades 10-12.

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