# Calculas Related Rates and Angle Problem

So i have recieve a problem in my calculas class that i have been working on for about 2 weeks and have come up empty handed in my attempt to find the answer, the question is quite lengthly, but is as follows:

a cruiser is steaming on a straight course at 20 knots. An airplane, flying so low that its angle of elevation from the boat can be neglected, is traveling at 200 knots along a striaght line making a 60 degree angle with the projected course of the cruiser. Both the plane and the ship are moving in the same general direction. A battery of guns aboard the cruiser is kept trained on the plane. At the instant the guns are pointed along a line perpendicular to the path of the ship, the plane is pne half nautical mile away. The guns will start firing at the plane at the instant it is nearest the ship. What is this minimum distance, and what angle are the guns making with the path of the ship at the instance they commence firing?

So i began by using the law of Sines and found the length of all the sides and then also by graphing it by hand i found the general region where the plane would be, then also was able to find that this region was .077 nMi on the path of the ship, also labeling the angle that i needed to find as theta, or however that is spelt in english, and labeling the other unknow angle as 120 - theta. I labeled the distance as z and using the law of sines, i had set z equal to sine(60)(.077)/sin(theta) and then figuring i will take the first derivative of this and set it equal to zero, but when i do this and solve it my angle i obtain is 89.99 degress which i kno is incorrect, i was wondering if you have any suggestions on what i do because it is due this upcoming friday and i would like to be able to do it correctly to obtain a decent mark this final quarter.

thank you very much

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StatusX
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jonmcclain said:
then also was able to find that this region was .077 nMi on the path of the ship,
I'm not sure where this number comes from.

You need to find the distance between the ship and the boat as a function of time. Do you know how to find the position of each as a function of time, and if so, how to find the distance between them? Once you do, minimize this by taking a derivative w.r.t. time and setting it equal to zero. It will probably be easier to find the square of the distance between them, since this will not have a square root and so will be easier to differentiatie, but has a minimum at the same point as the actual distance.

Try using the Law of Cosines to set up your equations for related rates. As StatusX said, set up your equations w.r.t time.

oh and i also was using the law of sins today when i worked on it, and will be able to find the theta angle i used the law of sines in setting up

sin(60)/z = sin(theta)/.077 z being the minimum distance wanted, then doing this i took the first dirivative(first set equal to z) and then also using the equation

sin(60)/z = sine(120-theta)/.283 this number being the y axis value, and then also taking the first dirivative of this(also set equal to z) and then attempted to use these 2 equations finding what a given variable might equal but i was un sucessful every time

so after this i decided that i didnt kno why iwas using those distances when they are changing in reality and im looking for a constant and those arnt the exact constant i need so i used the 20 knots and 10 knots and set up the same equation as my first listed and set up x as my distance(.077) and x being my 200 knots, when i took the dirivative i replaced x with 200 and ended up with theta and the first derivative of theta and z, so really im totally lost if any of you can help me i can scan a copy and if you would like we can chat, let me kno and that would be sweet, thankyou for your time