A boat in calm seas travels 13km east and 35 km north. Find the distance and direction of the trip, relative to to the north.

Pythagorean Theorem :

c ^ 2 = a ^ 2 + b ^ 2

c = sqrt ( a ^ 2 + b ^ 2 )

c = sqrt ( 13 ^ 2 + 35 ^ 2 )

c = sqrt ( 169 + 1225 )

c = sqrt ( 1394 )

c = 37.336 km

To solve this problem, we can use the Pythagorean theorem to find the distance traveled, and trigonometry to find the direction of the trip relative to the north.

First, we find the distance traveled using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the distance traveled is the hypotenuse, and the 13 km east and 35 km north form the other two sides of the right triangle. So, we have:

Distance^2 = (13 km)^2 + (35 km)^2

Simplifying:

Distance^2 = 169 km^2 + 1225 km^2

Distance^2 = 1394 km^2

Taking the square root of both sides:

Distance = √(1394 km^2)

Calculating the square root, we find:

Distance ≈ 37.34 km

Therefore, the distance of the trip is approximately 37.34 km.

Now, to find the direction of the trip relative to the north, we can use trigonometry. We have a right triangle where the eastward distance is the adjacent side and the northward distance is the opposite side. To find the angle θ (relative to the north), we can use the tangent function:

tan(θ) = Opposite / Adjacent

tan(θ) = 35 km / 13 km

Calculating the tangent:

θ ≈ tan^(-1)(35/13)

θ ≈ 68.02 degrees

Therefore, the direction of the trip relative to the north is approximately 68.02 degrees.