Question

A port and a radar station are 3 mi apart on a straight shore running east and west. A ship leaves the port at noon traveling at a rate of 15 mi/hr. If the ship maintains its speed and course, what is the rate of change of the tracking angle θ between the shore and the line between the radar station and the ship at 12:30 PM? (Use the Law of Sines)

Use the Law of Sines to find an equation relating the angle,

θ, the angle that the ship left the port at, the distance between the radar system and the ship, **a**, and the distance between the port and the ship, **s**. Evaluate any known trigonometric functions of 45° as needed. Use radians to express any other angles. (see image)

Step 1

Please look at thediagram on white board. I have used the diagram from the question itself and marked "a" on it.

As a first step, let's translate the information given in the question into relevant parameters.

- A port and a radar station are 3 mi apart on a straight shore running east and west. Hence, AB = 3 miles, and it will remain constant throughout.
- A ship leaves the port at noon traveling at a rate of 15 mi/hr. At12:30 PM (half an hour after start) the distance travelled by the ship = AC = s = Speed x time = 15 mi/hr x 0.5 hr = 7.5 mi/hr
- ds/dt = rate of change of s with time "t" = spped of the ship = 15 mi / hr

Step 2

In the triangle ABC, apply the Law of Sines. Please see the white board. This gives us a relationship between the three variables a, s and ϴ.

Step 3

Let's now differentiate this equation with respect to time "t". Recall that:

- Chain rule of different...

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