A batted baseball leaves the bat at an angle of 35.0 degrees above the horizontal and is caught by an outfielder 370 ft. from home plate at the same height from which it left the bat.

To find the initial velocity of the baseball, you can use the horizontal and vertical components of the motion.

Let's start by breaking down the given information:

Angle of launch (θ) = 35 degrees
Horizontal distance (d) = 370 ft
Vertical distance (h) = 0 ft (since it is caught at the same height from which it left the bat)

Now, we can find the horizontal and vertical components of the velocity:

Horizontal component:
The horizontal component of the velocity remains constant throughout the motion. We can use the formula:
v_horizontal = v_initial * cos(θ)

Vertical component:
The vertical component of the velocity changes due to the acceleration due to gravity. Therefore, we need to use the formula:
v_vertical = v_initial * sin(θ) - (g * t)

Since the ball is caught at the same height it left the bat, we know that the time taken (t) is the same for both the horizontal and vertical components.

Now, we need to find time (t):
Using the horizontal distance (d) and horizontal velocity (v_horizontal), we can rearrange the formula for distance (d = v * t) to solve for time (t):
t = d / v_horizontal

Now that we have the time, we can substitute it into the formula for the vertical component to find the initial velocity (v_initial):
v_vertical = 0 (since the ball falls to the same level it was launched)
0 = v_initial * sin(θ) - (g * t)
v_initial = (g * t) / sin(θ)

Where,
g = acceleration due to gravity (9.8 m/s^2 or 32.2 ft/s^2)

Now, Let's plug in the given values and calculate:

First, convert 370 ft to meters (1 ft = 0.3048 m):
d = 370 ft * 0.3048 m/ft = 112.776 m

Using the formula for time, we can calculate:
t = d / v_horizontal = 112.776 m / v_initial * cos(35)

Let's plug in all the values into the equation to find the initial velocity (v_initial):
v_initial = (9.8 m/s^2 * (112.776 m / v_initial * cos(35))) / sin(35)
Simplifying the equation,
v_initial = 9.8 m/s^2 * (112.776 m) / (v_initial * cos(35) / sin(35))

To solve this equation, we can make v_initial the subject:
v_initial * cos(35) / sin(35) = 9.8 m/s^2 * (112.776 m) / v_initial
v_initial^2 = 9.8 m/s^2 * (112.776 m) * sin(35) / cos(35)
v_initial^2 = 9.8 m/s^2 * (112.776 m) * tan(35)
v_initial^2 = (9.8 m/s^2 * 112.776 m) / cos(35)
v_initial = sqrt((9.8 m/s^2 * 112.776 m) / cos(35))

Calculating the value using a calculator, we find:
v_initial ≈ 43.2 m/s.

Therefore, the initial velocity of the baseball is approximately 43.2 m/s.

Incomplete.