Two boats depart simultaneously from a port P. Boat A travels along a straight path for 8\… — JAMB UTME

Two boats depart simultaneously from a port $P$ . Boat $A$ travels along a straight path for $8\text{ km}$ , while Boat $B$ travels along another straight path for $15\text{ km}$ . If the angle between their paths at the port is $60^\circ$ , what is the distance between the two boats in kilometers?

Answer: 13 km

Answer

The distance between the two boats is 13 km.

The distance between the two boats forms the third side of a triangle where two side lengths (

8\text{ km}

and

15\text{ km}

) and the included angle (

60^\circ

) are known. By the Cosine Rule,

d^2 = 8^2 + 15^2 - 2(8)(15)\cos(60^\circ) = 64 + 225 - 120 = 169

, so

d = \sqrt{169} = 13\text{ km}

Step-by-Step Solution

Formulate the geometric model

A triangle with two sides of length

8\text{ km}

and

15\text{ km}

, and an included angle of

60^\circ

The paths of the two boats and the distance between them form a triangle with two given side lengths and the included angle.

Set up the Cosine Rule formula

d^2 = 8^2 + 15^2 - 2(8)(15)\cos(60^\circ)

The Cosine Rule (

c^2 = a^2 + b^2 - 2ab\cos C

) is used when two sides and the included angle are known (SAS configuration).

Evaluate the trigonometric term and simplify

d^2 = 64 + 225 - 240(0.5) = 289 - 120 = 169

Since

\cos(60^\circ) = 0.5

, the subtraction term reduces to

120

Calculate the principal square root

d = 13

Taking the positive square root gives the distance in kilometers.

Key Concept

Applying the Cosine Rule to calculate the unknown side of a triangle given two sides and the included angle (SAS).

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