At noon, ship A is 100 Kilometers due east of ship B, Ship A is sailing west at 12 k/h, and ship B is sailing S10degrees west at 10k/h. At what time will the ships be nearest each other and what time will the distance be?
(not a right triangle)?
Not a right triangle?
You know angle, or SAS.
you want the other distance (S)
I think I would start with the law of cosines:
c^2=a^2+b^2 -2ab CosC
Ok. So c^2=100^2+b^2-2(100)b Cos100
or maybe this?
c^2=100^2-x+(-12t)^2-2(100-x)(-12t) Cos100
To find the time at which the ships will be closest to each other and the distance between them at that time, you can use the law of cosines.
The law of cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, we have:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this scenario, we are trying to find the distance between ship A and ship B (side c). We know that at noon, ship A is 100 kilometers due east of ship B. Ship A is sailing west at 12 km/h, and ship B is sailing S10 degrees west at 10 km/h.
To apply the law of cosines, we will first find the length of side a (distance traveled by ship A) and side b (distance traveled by ship B).
For ship A, the distance traveled can be calculated using the equation: distance = speed * time. So, the distance traveled by ship A is given by:
a = -12t, where t is the time in hours.
For ship B, we need to find the distance traveled in both the east and west direction. The eastward distance traveled is given by the equation: distance = speed * time. So, the eastward distance traveled by ship B is:
100 km - 10t, where t is the time in hours.
To find the total distance traveled by ship B (side b), we need to use the concept of bearings. The southward movement of S10 degrees west can be converted into east and west distance components. Using trigonometry, we can find the eastward and westward distances traveled by ship B as:
eastward distance = 10 * cos(10 degrees) = 9.848 km, (positive because it's eastward)
westward distance = 10 * sin(10 degrees) = 1.729 km, (positive because it's westward)
So, the westward distance traveled by ship B is 1.729 km - the westward distance traveled by ship B = 1.729 - (100 - 10t) = -98.271 + 10t.
Now, we can substitute the values of a, b, and C into the law of cosines equation:
c^2 = a^2 + b^2 - 2ab * cos(C)
c^2 = (-12t)^2 + (9.848 + (-98.271 + 10t))^2 - 2(-12t)(9.848 + (-98.271 + 10t)) * cos(100 degrees)
Simplifying this equation will give you a quadratic equation in terms of t. Solving this equation will provide you with the time (t) at which the ships will be closest to each other. Substitute this value of t into the distance equation a = -12t to find the distance between the ships at that time.
Note: The equation is quadratic, meaning you may find two solutions for t. One solution will represent the time at which the ships are nearest to each other, and the other solution will represent a point in time where the ships move apart from each other after being closest.