The target is to be dropped from the point

A at t = 0, the same moment as the gun is
fired. The bullet hits the target at a point
P. Let the initial speed of the bullet be v0 =
106 m/s, let the angle between the vector v0
and the horizontal (x) direction be θ = 57.3
and let AB = 91 m. The distance d = OB is
the x-coordinate of the target.
The acceleration of gravity is 9.8 m/s^2

The height BP where the collision takes
place is

i know i have to find the x component and y component in this equation

106.3 cos 57.3
106.3 sin 57.3

then what do I do?

Some clarification of the meaning and location points A, B, P, and O are needed.

To find the height BP where the collision takes place, you are correct that you need to calculate the x-component and y-component of the bullet's initial velocity.

The x-component of the velocity (Vx) can be calculated using the formula:
Vx = v0 * cos(theta)
where v0 is the initial speed of the bullet and theta is the angle between the vector v0 and the horizontal direction.

Substituting the given values:
Vx = 106.3 m/s * cos(57.3°)

Similarly, the y-component of the velocity (Vy) can be calculated using the formula:
Vy = v0 * sin(theta)
where sin(theta) is the ratio of the length of the opposite side to the hypotenuse.

Substituting the given values:
Vy = 106.3 m/s * sin(57.3°)

Now, to find the height BP, you need to calculate the time it takes for the bullet to reach the target point P horizontally. This can be done using the formula:
t = d / Vx
where d is the horizontal distance between the point A and the target point P.

Substituting the given value:
t = 91 m / Vx

Now, you can calculate the height BP using the formula:
BP = Vy * t - (0.5 * g * t^2)
where g is the acceleration due to gravity.

Substituting the values:
BP = (106.3 m/s * sin(57.3°)) * (91 m / (106.3 m/s * cos(57.3°))) - (0.5 * 9.8 m/s^2 * (91 m / (106.3 m/s * cos(57.3°))))^2

By evaluating this expression, you will find the height BP where the collision takes place.