menu
bartleby
search
close search
Hit Return to see all results

NortheastNcourse45°ARadar stationPortB3Z

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)
Northeast
N
course
45°
A
Radar station
Port
B3
Z
help_outline

Image Transcriptionclose

Northeast N course 45° A Radar station Port B3 Z

fullscreen
check_circleAnswer
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
Northeast
N
course
a
45°
A
I B
Radar station
Port
Z
help_outline

Image Transcriptionclose

Northeast N course a 45° A I B Radar station Port Z

fullscreen
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.

a
Sin A
Sin B
a
Sin( 4) Sin0
2aSin0 = s
help_outline

Image Transcriptionclose

a Sin A Sin B a Sin( 4) Sin0 2aSin0 = s

fullscreen
Step 3

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

  • Chain rule of different...
2aCosa2Sine da _ ds
=15
dt
dt
dt
help_outline

Image Transcriptionclose

2aCosa2Sine da _ ds =15 dt dt dt

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Our solutions are written by experts, many with advanced degrees, and available 24/7

See Solution
Tagged in

Math

Calculus

Derivative

Sorry about that. What wasn’t helpful?