I need a plan to solve this problem and I'm stuck. Please help.

A basketball laeaves a player's hands at a height of 2.10m above the floor. The basket is 2.60m above the floor. The player shoots the ball at a 38.5 degree angle. If the shot is made from a horizontal distance of 12.0m and must be accurate to +or- 0.22 (horizontally), what is the range of initial speeds allowed to make the basket?

Derive the formula for horizontal "range" distance R travelled between the two heights, as a function of V. Then find the values of V that correspond to 12.22 and 11.78 m range for R.

R = V cos 38.5 * T , where T is the time of flight between the two heights.

Assume the ball must be coming down, not going up, when it reaches 2.6 m height.

Y = 2.1 + V sin 38.5 T - 4.9 T^2 = 2.6
4.9 T^2 - 0.623 VT + 0.5 = 0
T = [0.623 + sqrt(0.388V^2 - 9.8)]/9.8
I have not used the smaller of the two roots of the quadratic equation because that corresponds to the ball going UP.

The horizontal distance travelled in time T is
R = V cos 38.5 T = 0.783 VT
R = 0.0799V[0.623 + sqrt(0.388V^2- 9.8)]

Check my math and thinking. You may have to use graphical or iterative means to find the value of V that correspond to R = 12.22 and 11.78 m

To solve this problem, we need to find the range of initial speeds that will allow the basketball to reach the basket. We can use the following steps to solve the problem:

Step 1: Break down the problem
- The problem involves finding the range of initial speeds of the basketball to make the basket.
- The basketball is shot at an angle of 38.5 degrees.
- The basketball leaves the player's hands at a height of 2.10m above the floor.
- The basket is located 2.60m above the floor.
- The horizontal distance between the player and the basket is 12.0m.
- The shot must be accurate within a range of +/- 0.22m horizontally.

Step 2: Analyze the problem
- We need to determine the initial speed at which the basketball should be shot to reach the basket.
- Since the initial speed is unknown, we can use formulas and equations of motion to solve the problem.

Step 3: Identify the relevant equations
- We can use the equations of projectile motion to solve this problem.
- The key equations are:
- Horizontal distance (range) = initial velocity x time of flight x cos(angle)
- Vertical distance = initial velocity x time of flight x sin(angle) - 0.5 x g x (time of flight)^2
- Time of flight = 2 x (initial velocity x sin(angle)) / g

Step 4: Solve the problem
- We will calculate the range of initial speeds that will allow the basketball to reach the basket within the specified accuracy.
- First, we calculate the time of flight:
- Substitute the known values into the equation: time of flight = 2 x (initial velocity x sin(angle)) / g
- Use the known value of the angle (38.5 degrees) and the acceleration due to gravity (9.8 m/s^2)
- Rearrange the equation to solve for the initial velocity: initial velocity = (g x time of flight) / (2 x sin(angle))

- Next, we can substitute the calculated value of the initial velocity into the range equation:
- Substitute the known values (range = 12.0m, angle = 38.5 degrees, and initial velocity from the previous step) into the equation: horizontal distance = initial velocity x time of flight x cos(angle)
- Rearrange the equation to solve for the initial velocity: initial velocity = horizontal distance / (time of flight x cos(angle))

- Finally, we can calculate the range of initial speeds allowed to make the basket:
- Substitute the known values into the equation: range = initial velocity x time of flight x cos(angle)
- Rearrange the equation to solve for the initial velocity: initial velocity = range / (time of flight x cos(angle))

Step 5: Calculate the numerical solution
- Substitute the known values (range = 12.0m, angle = 38.5 degrees, and initial velocity from the previous step) into the equation: initial velocity = range / (time of flight x cos(angle))
- Calculate the initial velocity using the calculated time of flight and angle.

Step 6: Calculate the allowed range of initial velocities
- Calculate the difference between the maximum allowed initial velocity (initial velocity + 0.22m) and the minimum allowed initial velocity (initial velocity - 0.22m).

By following these steps, you should be able to determine the range of initial speeds allowed to make the basket.