grade 11 applied math

Question

 Alex, Curtis and Michael go boating together on a calm day. They leave the marina dock and travel at a speed of 40 km/h due east for 3 hours. Then, they travel 22° west of south for 2 hours at a speed of 30 km/h. At that point, their boat breaks down and they call the mechanic at the marina dock to come and help them.

To get to them in the shortest time, how far must the mechanic travel and in what direction?

1 mark for an appropriate diagram
1 mark for finding the enclosed angle
1 mark for setting cosine law up properly
1 mark for finding the distances the boats travel
1 mark for finding the measure of angle A
1 mark for giving a correct direction

Answer 1

Starting at the origin

 

THey go 40 km/hr x 3 hours = 120 km due east ---> (120,0)

 

Next 22 degrees west of south is 68 degrees below the positive x-axis.

The right triangle formed therefore has a 68 degree angle at (120,0)

THe hypotenuse is 30 km/hr x 2 hours = 60 km.

 

sin 68 = y/60  where y is the other leg of the right triangle (vertical distance from the positive x-axis, downwards)

y = 60 * sin 68 = 55.63

The leg of the right triangle is then x^2 + y^2 = 60^2

                                                   x =  sqrt(3600 - y^2) = 22.4789

Also this leg can be found as:

    cos 68 = x / 60 ---> x = 60 * cos 68 = 22.476395605

 

Average these gives an estimate of 22.47765

 

So they are at (120 - 22.47765 , 0 - 55.63)

 

                        = (97.52235, -55.63)

 

when the boat breaks down

 

By pythagorean ,

The mechanic goes 112.273 = sqrt ( 97.52235^2 + (-55.63)^2 ) 

The tangent of the angle is -55.63/97.52235 = 29.7018767 degrees south of east