a boat travels 60 km due east. it then adjusts its course by 25 degrees northward and travels another 90 km in the new direction. how far is the boat from its initial position to the nearest kilometer?

East distance = 60 + 90 cos 25 = 141.6

North distance = 90 sin 25= 38

d^2 = 141.6^2 + 38^2
d = 147

To find the distance the boat is from its initial position, we can use the concept of vector addition. Let's break down the boat's motion into two components: eastward and northward.

First, let's calculate the eastward component. Since the boat travels 60 km due east, the distance in the eastward direction is simply 60 km.

Next, let's calculate the northward component. To do this, we need to find the component of the 90 km distance in the new direction (25 degrees northward) that is perpendicular to the eastward component we calculated earlier.

To find this perpendicular component, we multiply the total distance (90 km) by the sine of the angle between the northward direction and the eastward direction (25 degrees). Mathematically, this can be written as:

Perpendicular Component = 90 km * sin(25°)

Using a scientific calculator, we find that the Perpendicular Component is approximately 37.835 km.

Now, we can find the total displacement of the boat from its initial position by adding the eastward component and the northward component. Mathematically, this can be written as:

Total Displacement = Eastward Component + Perpendicular Component

Total Displacement = 60 km + 37.835 km

Total Displacement = 97.835 km

Therefore, the boat is approximately 97.835 km away from its initial position to the nearest kilometer.