The 3-point line in basketball is 23 ft 9 in away from the point on the floor vertically below the center of an 18-in diameter rim. The rim is 10 ft above the floor. Kim shoots a three pointer using a 9-in diameter ball from the 3-point line that just clears the basket. The release height was 8 ft. What is the smallest release angle, measured relative to horizontal?

To find the smallest release angle, we can break down the problem into two separate parts: finding the horizontal distance covered by the ball and finding the vertical height reached by the ball. Once we have those values, we can use trigonometry to calculate the angle.

Let's start by calculating the horizontal distance covered by the ball. We know that the 3-point line is 23 ft 9 in away from the point vertically below the center of the rim. Since the rim is 10 ft above the floor, we can calculate:

Horizontal distance = 23 ft 9 in

To simplify the calculations, let's convert the measurements to a consistent unit, such as inches:

Horizontal distance = 23 ft * 12 in/ft + 9 in = 276 in + 9 in = 285 in

Now, let's calculate the vertical height reached by the ball. We know that the release height was 8 ft, and the ball just cleared the 10 ft high rim. Therefore:

Vertical height = 10 ft - 8 ft = 2 ft

To calculate the smallest release angle, we can use trigonometry. The release angle is the angle between the horizontal distance and the hypotenuse formed by the horizontal and vertical distances. Since we have the values for the horizontal distance and the vertical height, we can use the tangent function to find the angle:

tan(angle) = vertical height / horizontal distance

angle = arctan(vertical height / horizontal distance)

angle = arctan(2 ft / 285 in)

Using a scientific calculator or an online trigonometry calculator, we can find the arctan of (2 / 285) to get the value of the angle. The angle will be measured in radians or degrees, depending on the calculator used.

Therefore, the smallest release angle, measured relative to the horizontal, can be calculated using the given information and the arctan function.