Two boats leave whitby harbour at the same time one boat heads 18 km to it's destination and the other boat heads 11 km to its destination. If the boats headed on a course 70 degrees apart, what is the distance between the boats ?

use the law of cosines:

d^2 = 18^2 + 11^2 - 2(18)(11)cos 70°

To find the distance between the two boats, we can use the Law of Cosines. This formula relates the lengths of the sides of a triangle to the cosine of one of its angles.

In this case, we have a triangle with sides of lengths 18 km, 11 km, and an unknown distance between the boats. We also know that the angle opposite to the unknown distance is 70 degrees.

The Law of Cosines states that:

c^2 = a^2 + b^2 - 2ab * cos(C),

where c is the unknown distance between the boats, a and b are the lengths of the known sides, and C is the angle between the known sides.

Let's substitute the known values into the formula:

c^2 = 18^2 + 11^2 - 2(18)(11) * cos(70).

Calculating this expression:

c^2 = 324 + 121 - 396 * cos(70).

Using a calculator, we find that cos(70) is approximately 0.342.

c^2 = 324 + 121 - 396 * 0.342,
c^2 ≈ 324 + 121 - 135.672,
c^2 ≈ 309.328.

Finally, we take the square root of both sides to find the value of c:

c ≈ √309.328,
c ≈ 17.6 km (rounded to one decimal place).

Therefore, the distance between the two boats is approximately 17.6 km.