Solutions

Solution

Given Information: The first distance (A) from Camp Tana to Home Base at Mananara is on a bearing of N 39 E, is 400 km long, and you will be traveling at 220km/hr. The second distance (T), from Home Base to Tolanaro is 1000km on a bearing of S 199 W and at a speed of 40 km/hr. The last speed from Tolanaro back to Camp Tana is 92km/hr.

Step One: Find the measure of angle “M”.

    -         Since the bearing is 199 degrees, you subtract 180 from it because you are
            going south.
    -         That gives you 19 degrees, which is angle “Q”.
    -         Back at Camp Tana, the bearing is 39 degrees. Because you are moving
          northeast, you subtract 39 from
          90 degrees.
    -         This gives you 51 degrees, which is angle “T”.
    -         By using Alternate Interior Angles, you know that angle “B” must also equal
          51 degrees.
    -         Now you can add 19 and 51, and subtract the total from 90, to get angle “M”.
    -         Angle “M” is 20 degrees.

Step Two: Find the length of side “C”.
-         Now that we have the measure of angle “M”, we can use the Law of Cosines to find the length of “C”.
-         Law of Cosines= c²= a² + b²– (2abcos C)
-         c²= (400)²+ (1000)²– (2(400)(1000)cos 20)

- The length of c, therefore, is 638.94 km.

Step Three: Find angle “A” in order to find course bearing for trip home.
    -         Use Law of Sines: SinM/m = SinA/a
    -         Sin 20/638.94 = Sin A/400
    -         (400 Sin(20)/638.94) = .214…
    -         Sin ^-1(.214…) = 12.36

- So the angle measure of “A” is 12.36 degrees, and the course bearing is N 12.36 E

Step Four: Find the time it will take to complete the mission.
    -         Use the Distance Formula: D = RT, T = D/R
    -         Find the time it will take to go from point T to point M (distance A):
            - Rate: 220 km/hr         Distance: 400 km
            - T = 400/220 = 1.8 hours

- Find the time it will take to go from point M to point A (distance T):

            - Rate: 40 km/hr           Distance: 1000 km
            - T = 1000/40 = 25 hours
         -    Find the time it will take to go from point A to point T (distance C):
            - Rate: 92 km/hr           Distance: 638.94 km
            - T = 638.94/92 = 6.945 hours
    -    Next, convert the answers into hours and minutes by multiplying the
        numbers to the right of the decimal by 60.

            - 1.8 = 1 hour and .8(60) = 1 hour and 48 minutes
            - 25 = 25 hours
            - 6.945 = 6 hours and .945(60) = 56.7 = 56 minutes and .7(60) = 1 hour 56 minutes and 42 seconds
            - 5 hours for the break period
     -    Finally, add up all the times to see if you can complete the mission before 2300 hours.
            - 1 hour and 48 minutes + 25 hours + 1 hour 56 minutes and 42 seconds + 5 hours   = 38 hours 44
              minutes and 42 seconds

-You will arrive at 1900 hours with 4 hours to spare!

- You can successfully complete this mission!!!