A ship sets out to sail to a point 174 km due north. An unexpected storm blows the ship to a point 132 km due east of its starting point. How far must it now sail to reach its original destination? In what direction must it now sail to reach its original destination?

It must sail 174 km north and 132 km west. Take the hypotenuse for the distance required. The direction is that of the hypotenuse. (arctan 132/174 West of North)

To find the distance the ship must now sail to reach its original destination, we can use the Pythagorean theorem. The ship has been blown 132 km due east and was originally supposed to travel 174 km due north. This forms a right-angled triangle.

Let's call the distance the ship needs to sail "d". The distance the ship traveled to the east is 132 km, and the distance it was supposed to travel to the north is 174 km. So, according to the Pythagorean theorem:

d^2 = 132^2 + 174^2

To calculate this, we square the values for 132 and 174, sum them, and then take the square root of the result to find "d".

d^2 = 174^2 + 132^2
d^2 = 30,276 + 17,424
d^2 = 47,700

Taking the square root of both sides:

sqrt(d^2) = sqrt(47,700)
d = √47,700
d ≈ 218.1 km

Therefore, the ship must now sail approximately 218.1 km to reach its original destination.

To determine the direction the ship must now sail, we can use trigonometry. We have a right-angled triangle, and we know the lengths of the two sides adjacent to the angle we want to calculate.

We can use the inverse tangent (tan^(-1)) function to determine the angle. Using the lengths of the sides, we have:

tan(theta) = opposite/adjacent
tan(theta) = 174/132

To find theta, we take the inverse tangent of both sides:

theta = tan^(-1)(174/132)
theta ≈ 50.5 degrees

Therefore, the ship must now sail approximately 218.1 km in a direction approximately 50.5 degrees (measured clockwise from due east) to reach its original destination.