) An object is moving on a straight line which is 15 centimeters away from the center of a circle of radius 28 centimeters. (Both the circle and the straight line are on the same plane.) A source of light is located on the line drawn from the center of the circle perpendicular to the original line, and is 10 centimeters away from the original line. What is the speed of the shadow of the moving object (projected on the circle) when the object is one centimeter away from the intersection of the two lines?

is the light source between the center of the circle and the object's line, or on the far side of the line? That will affect where the shadow falls.

Also, you have not indicated the speed of the object, dx/dt.

In either case, label the light source L, the object P and the center of the circle O. Let the shadow fall on the circle at point S. You want to figure angle LOS = LOP = θ.

Once you have figured dθ/dt in terms of dx/dt (when x=1), then the speed along the circle can be found using

s = rθ
ds/dt = r dθ/dt

To find the speed of the shadow of the moving object at a specific point, we need to understand the geometry of the situation.

Let's break down the problem step by step:

Step 1: Visualize the situation
- Draw a circle with a radius of 28 centimeters, representing the center of the circle.
- Mark a straight line, 15 centimeters away from the center of the circle.
- Draw a line perpendicular to the original line, passing through the center of the circle.
- Place a light source on the perpendicular line, 10 centimeters away from the original line.
- Imagine an object moving along the original line.

Step 2: Find the point of intersection of the perpendicular line and the original line
- Since the perpendicular line goes through the center of the circle, it will intersect the original line at a point.
- Let's call this point P.

Step 3: Measure the distance from the light source to point P
- The distance from the light source to point P is given as 10 centimeters.

Step 4: Measure the distance from point P to the moving object
- As the object moves along the original line, measure the distance from point P to the object.
- Let's call this distance d. Initially, d is 1 centimeter.

Step 5: Define a reference point on the circle
- Select a point on the circle as a reference point.
- Let's call this point Q.

To find the speed of the shadow of the moving object projected on the circle, we need to calculate the rate at which the shadow moves along the circumference when the object is 1 centimeter away from point P.

Step 6: Calculate the distance travelled by the shadow
- The distance travelled by the shadow is equal to the length of the arc on the circle from the reference point Q to the current position of the shadow.

Step 7: Calculate the length of the arc
- Use the formula for the length of a circular arc:
Arc length = angle (in radians) x radius

Step 8: Calculate the angle (in radians)
- The angle between the reference point Q, the center of the circle, and the current position of the shadow is equal to the angle at the center of the circle between the reference point Q, the center of the circle, and the current position of the shadow.
- To find this angle, we can use trigonometry.
- The tangent of this angle is equal to the opposite side (d in our case) divided by the adjacent side (10 centimeters).
- So, tangent(angle) = d/10
- Using inverse tangent, we can find the angle: angle = atan(d/10)

Step 9: Calculate the length of the arc
- Plug the calculated angle (in radians) and the radius of the circle (28 centimeters) into the formula for the length of a circular arc:
Arc length = angle x radius

Step 10: Calculate the speed of the shadow
- The speed of the shadow is equal to the rate at which the shadow moves along the circumference when the object is 1 centimeter away from point P.
- To calculate this speed, divide the length of the arc by the time it takes for the object to travel 1 centimeter.
- Let's assume the time taken is 1 second.
- Speed of the shadow = Arc length / Time taken

By following these steps, you can calculate the speed of the shadow of the moving object when it is one centimeter away from the intersection of the two lines.