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Part A: To write each of the vectors in trigonometric form, we need to find their horizontal and vertical components.
For the velocity vector:
Horizontal component = 60 mph * cos(130) = -25.40 mph
Vertical component = 60 mph * sin(130) = 56.05 mph
Therefore, the velocity vector in trigonometric form is:
v = -25.40i + 56.05j mph
For the wind vector:
Horizontal component = 8 mph * cos(230) = -4.15 mph
Vertical component = 8 mph * sin(230) = -6.56 mph
Therefore, the wind vector in trigonometric form is:
w = -4.15i - 6.56j mph
Part B: To find the sum of the two vectors, we simply add their horizontal and vertical components:
v + w = (-25.40 - 4.15)i + (56.05 - 6.56)j mph
Simplifying, we get:
v + w = -29.55i + 49.49j mph
Therefore, the sum of the two vectors in trigonometric form is:
v + w = -29.55i + 49.49j mph
Part C: To find the true speed of the ball, we can use the Pythagorean theorem:
True speed = sqrt((-29.55)^2 + (49.49)^2) mph
Simplifying, we get:
True speed = 57.59 mph
Therefore, the true speed of the ball is 57.59 mph.
Part D: To find the true direction of the ball, we can use the inverse tangent function:
True direction = arctan(49.49/-29.55) degrees
Simplifying, we get:
True direction = -59.52 degrees
However, since the ball was kicked in the direction of 130 degrees relative to the air, we need to add this value to the true direction to find the final direction of the ball:
Final direction = -59.52 + 130 = 70.48 degrees
Therefore, the true direction of the ball is 70.48 degrees.