To determine the direction in degrees that the sling is pointing when the rock is released, we need to consider the position of the rock and the tangent line to the circle at that point.
Since the broomstick is 15 feet due east of you, we know that the rock needs to be released in a direction that will make it travel eastward. This means that the tangent line to the circle at the point of release should be pointing due east.
To find this tangent line, we can consider the geometry of the situation. The radius of the circle is 3 feet, and the rock is being swung around your head. When you release the rock, it will continue to move along the tangent line to the circle through its position at the time of release.
Since the rock is 15 feet due east of you, we can imagine a line connecting you, the center of the circle, to the point where the rock is released. This line represents the radius of the circle. Now, draw a perpendicular line to this radius at the point of release. This perpendicular line represents the tangent line to the circle.
Next, we need to determine the angle between the radius line connecting the center of the circle to the point of release and the line pointing due east. This angle will give us the direction in degrees that the sling is pointing.
To find this angle, we can use inverse trigonometric functions. The cosine function can help us find the angle since it deals with adjacent sides and the hypotenuse. The adjacent side in this case is the length of the radius of the circle (3 feet), and the hypotenuse is the distance between you and the point of release (15 feet).
Using the formula for the cosine of an angle:
cos(angle) = adjacent / hypotenuse
we can rearrange the formula to solve for the angle:
angle = arccos(adjacent / hypotenuse)
Plugging in the values we have:
angle = arccos(3 / 15)
Calculating this using a calculator, the angle is approximately 77.57 degrees.
Therefore, when you release the rock, the sling is pointing in the direction of approximately 77.57 degrees clockwise from due north.