Mathematical Modeling: Sports and Athletics

Average velocity is the total distance traveled divided by the total time taken. In this case, the total distance is 100 meters and the total time is 10 seconds. Therefore, the average velocity is 100 meters / 10 seconds = 10 m/s.

The maximum height reached by the ball can be calculated using the formula: h = (v^2 * sin^2(theta)) / (2 * g), where v is the initial velocity, theta is the angle of projection, and g is the acceleration due to gravity. Plugging in the values, we get: h = (20^2 * sin^2(45)) / (2 * 9.81) = 15 meters.

The maximum distance the ball will travel can be calculated using the formula: d = (v^2 * sin(2 * theta)) / g, where v is the initial velocity, theta is the angle of projection, and g is the acceleration due to gravity. Plugging in the values, we get: d = (30^2 * sin(2 * 30)) / 9.81 = 55 meters.

The angle at which the player should shoot the ball to make the shot can be calculated using the formula: theta = arctan(h / d), where h is the height of the basket and d is the distance from the basket. Plugging in the values, we get: theta = arctan(10 / 15) = 45 degrees.

The maximum height reached by the ball can be calculated using the formula: h = (v^2 * sin^2(theta)) / (2 * g), where v is the initial velocity, theta is the angle of projection, and g is the acceleration due to gravity. Since the ball is hit horizontally, the angle of projection is 0 degrees. Plugging in the values, we get: h = (120^2 * sin^2(0)) / (2 * 9.81) = 20 meters.

The maximum distance the ball will travel can be calculated using the formula: d = (v^2 * sin(2 * theta)) / g, where v is the initial velocity, theta is the angle of projection, and g is the acceleration due to gravity. Plugging in the values, we get: d = (60^2 * sin(2 * 45)) / 9.81 = 180 meters.

The total distance covered by the cyclist can be calculated using the formula: distance = speed * time. Plugging in the values, we get: distance = 30 km/h * 2 hours = 60 km.

The total distance of the marathon can be calculated using the formula: distance = speed * time. First, we need to convert the runner's time to hours: 3 hours and 30 minutes = 3.5 hours. Then, we can plug in the values: distance = 10 km/h * 3.5 hours = 42 km.

The swimmer's average speed can be calculated using the formula: speed = distance / time. First, we need to convert the swimmer's time to seconds: 20 minutes = 1200 seconds. Then, we can plug in the values: speed = 1500 meters / 1200 seconds = 1.25 m/s.

The maximum height reached by the ball can be calculated using the formula: h = (v^2 * sin^2(theta)) / (2 * g), where v is the initial velocity, theta is the angle of projection, and g is the acceleration due to gravity. Plugging in the values, we get: h = (25^2 * sin^2(60)) / (2 * 9.81) = 17.5 meters.

The maximum distance the ball will travel can be calculated using the formula: d = (v^2 * sin(2 * theta)) / g, where v is the initial velocity, theta is the angle of projection, and g is the acceleration due to gravity. Plugging in the values, we get: d = (35^2 * sin(2 * 45)) / 9.81 = 67.5 meters.

The angle at which the player should shoot the ball to make the shot can be calculated using the formula: theta = arctan(h / d), where h is the height of the basket and d is the distance from the basket. Plugging in the values, we get: theta = arctan(10 / 18) = 45 degrees.

The maximum height reached by the ball can be calculated using the formula: h = (v^2 * sin^2(theta)) / (2 * g), where v is the initial velocity, theta is the angle of projection, and g is the acceleration due to gravity. Since the ball is hit horizontally, the angle of projection is 0 degrees. Plugging in the values, we get: h = (130^2 * sin^2(0)) / (2 * 9.81) = 17.5 meters.

The maximum distance the ball will travel can be calculated using the formula: d = (v^2 * sin(2 * theta)) / g, where v is the initial velocity, theta is the angle of projection, and g is the acceleration due to gravity. Plugging in the values, we get: d = (70^2 * sin(2 * 30)) / 9.81 = 180 meters.