A boat in calm seas travels 19 km east and 32 km south. Find the distance and direction of the trip, relative to the south. What is the distance? What is the direction?
I'm confused about how to solve this problem. Could someone show me the steps?
to get the resultant or the total distance, we use distance formula:
d = sqrt[ (x2-x1)^2 + (y2-y1)^2 ]
where
(x1,y1) is our reference/starting point, and
(x2,y2) is the final point
assuming our reference point is at the origin which is at (0,0)
d = sqrt[ (x2)^2 + (y2)^2 ]
substituting:
d = sqrt[ (19)^2 + (-32)^2 ]
d = 37.22 km
*note: we use -32 since it is travelling south/downward direction
getting the direction: let theta = angle from x-axis measured counterclockwise,,
tan (theta) = y/x
tan (theta) = (-32)/19
theta = -59.3 degrees
*note: the negative sign means opposite direction, or it is rotated clockwise
since we need the angle to be relative to south/y-axis,
90 - 59.3 = 30.7 degrees from south
thus distance = 37.22 km , 30.7 degrees from south
hope this helps~ :)
Thank you so much! This definitely helped! =)
To solve this problem, you can use the Pythagorean theorem to find the distance and basic trigonometry to find the direction.
Step 1: Draw a diagram
Start by drawing a diagram to represent the boat's path. Mark the starting point and label the distances traveled east and south.
Step 2: Calculate the hypotenuse
Using the Pythagorean theorem, we can find the distance of the trip (referred to as the hypotenuse) by finding the square root of the sum of the squares of the distances traveled east and south.
Distance = √(19^2 + 32^2)
Distance = √(361 + 1024)
Distance = √1385
Distance ≈ 37.19 km
So, the distance of the trip is approximately 37.19 km.
Step 3: Calculate the direction
To find the direction, we can use trigonometry. In this case, we will use the tangent function.
Direction = atan(32/19)
Direction ≈ 58.49°
Therefore, the direction of the trip, relative to the south, is approximately 58.49°.
In summary, the distance of the trip is approximately 37.19 km, and the direction relative to the south is approximately 58.49°.