Projectile Motion reduced to trig problem

[SEZHUR]

Homework Statement

THIS PROBLEM HAS BEEN SOLVED

Quoted verbatim: A ship has maneuvered to within 2500m of an 1800m high mountainous island, as shown below. If the shoreline on the other side of the island is horizontally 300m from the base, what are the distances from the shore at which another ship can be safe from the guns of the first ship?
[Also, the ships gun fires at 250m/s]

Homework Equations

The Attempt at a Solution

What i figure is that we need to find the two angles that will hit the island at it's apex, discern which of the two arc's has it's apex before the island (so the shot will land as close to the island as possible), find out where that arc lands and subtract 2800 for the final answer. The problem comes in when i have to solve for the angle (my trig's not so good).

First i find time in terms of the range and initial velocity:
d=vt
2500=250cos(x)t
t=10/cos(x)

then substitute into the equation for the height

d=vt+(at^2)/2
1800=(250sin(x))(10/cos(x)) - 4.9(10/cos(x))^2
2500tan(x) - 490/cos(x)^2 - 1800 = 0

This is where i get stuck. I've already posted just this last part on a math forum to no avail, so I'm trying here now because I'm beginning to grow suspicious of my method. It seems to me that the trig question is too hard (based on the delay in response from the math forum) for the 100 level paper I'm doing.
Any help is welcome.
Thanks

 

[anathema]

Thank you for posting your problem here. I would like to offer some suggestions and clarifications for solving this problem.

Firstly, you are on the right track by trying to find the angle at which the ship's gun will hit the island at its apex. This angle can be found by using the inverse tangent function (tan^-1) and the known dimensions of the island (1800m height and 300m horizontal distance from the base).

Once you have found this angle, you can use basic trigonometry to find the horizontal distance at which the ship's gun will hit the island. This can be done by using the tangent function (tan) and the known height (1800m) and angle (found in the previous step).

Now, to find the distance from the shore at which another ship can be safe from the guns of the first ship, we need to take into account the initial distance between the first ship and the island (2500m). This means that the second ship needs to be at least 2500m + (horizontal distance found in previous step) away from the island to be safe from the first ship's guns.

I hope this helps and clarifies the steps needed to solve this problem. If you have any further questions or need any additional assistance, please do not hesitate to ask. Good luck with your paper!