A ferry transports tourists among three islands. It sails from the first island

Question

A ferry transports tourists among three islands. It sails from the first island to the second island, 4.90 km away, in a direction 37.0° north of east. It then sails from the second island to the third island in a direction 74.5° west of north. Finally, it returns to the first island, sailing in a direction 28.0° east of south. Hints: Make a sketch illustrating the vector addition. The magnitudes of the second and third displacements are unknown. Call them b and c. You should end up with two linear equations in the two unknowns; solve the system of equations.

(a) Calculate the distance between the second and third islands.

(b) Calculate the distance between the first and third islands.

Explanation / Answer

Here,

let's assume that the first island is origin

and east is the positive x direction

north is the positive y direction

a) for the second island

d2 = 4.90 * (cos(37) i + j * sin(37))

d23 = b * (- sin(74.5) i + j * sin(74.5) i)

d31 = c * (sin(28) i - j * cos(28))

NOw, as d2 + d23 + d31 = 0

4.90 * (cos(37) i + j * sin(37)) + b * (- sin(74.5) i + j * sin(74.5) i) + c * (sin(28) i - j * cos(28))

4.90 * cos(37 degree) - b * sin(74.5 degree) + c * sin(28 degree) = 0 ----(1)

and

4.90 * sin(37 degree) + b * cos(74.5 degree) - c * cos(28 degree) = 0 ---(2)

solving 1 and 2

b = 6.67 km

c = 5.35 km

disance between second and third island is 6.67 km
b) distance between first and third island is 5.35 km

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