The bases on a baseball diamond are 27.4m apart. The pitcher pitches,and the batter hits a fly ball straight up 15m. What is the maximum angle of elevation of the ball to the nearest degree, as seen by the pitcher if he is standing at the center of the diamond?
HELP
To find the maximum angle of elevation of the ball as seen by the pitcher, we can use trigonometry. We can consider the triangle formed by the pitcher, the point where the ball was hit, and the highest point of the ball's trajectory.
In this triangle, the distance from the pitcher to the point where the ball was hit is half of the distance between the bases, which is 27.4m / 2 = 13.7m. The height of the triangle is the maximum height reached by the ball, which is 15m.
Now we can use the trigonometric function tangent to find the angle of elevation. The tangent of an angle is equal to the opposite side divided by the adjacent side.
In this case, the opposite side is the height of the triangle (15m) and the adjacent side is the distance from the pitcher to the point where the ball was hit (13.7m).
Therefore, the tangent of the angle of elevation is equal to 15m / 13.7m.
Tangent(angle) = 15m / 13.7m
To find the angle itself, we can take the inverse tangent (arctan) of both sides:
Angle = arctan(15m / 13.7m)
Using a scientific calculator or an online trigonometric calculator, we find that the angle is approximately 49.4 degrees.
Therefore, the maximum angle of elevation of the ball, as seen by the pitcher, is about 49 degrees.
To find the maximum angle of elevation of the ball as seen by the pitcher, we can use trigonometry. We can create a right triangle using the distance between the pitcher and the landing spot of the ball, with the distance between the bases as the base of the triangle, and the height of the ball as the height of the triangle.
1. First, we need to find the distance between the pitcher and the landing spot of the ball. Since the bases are 27.4m apart, the distance from the pitcher to the landing spot is half of this distance, which is 27.4m/2 = 13.7m.
2. Now that we have the base and the height of the right triangle, we can use the inverse tangent function (tan^⁻1) to find the angle of elevation. The formula is:
angle = tan^⁻1(height / base)
The height of the triangle is 15m, and the base is 13.7m.
angle = tan^⁻1(15 / 13.7)
3. Now, we can use a scientific calculator or an online calculator to find the inverse tangent of 15 / 13.7. This will give us the angle of elevation.
Using the calculator, we find that the angle is approximately 47.18 degrees.
Therefore, the maximum angle of elevation of the ball as seen by the pitcher, when standing at the center of the diamond, is approximately 47 degrees to the nearest degree.
distance to the plate: 27sqrt2 /2
= 27.4/2 sqrt2== 19.3718
theta=arc tan 15/19.37=37.75 degrees
27/sqrt2 = 19.1 from home to mound
tan a = 15/19.1