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?
X = hor = 19km.
Y = ver = -32km.
d^2 = X^2 + Y^2,
d^2 = (19)^2 + (-32)^2,
d^2 = 361 + 1024 = 1385,
d = 37.2km.
tanA = Y/X = -32/19 = -1.6842,
A = -59.3deg,CW=360 - 59.3 = 300.7deg CCW=300.7 - 270=30.7deg East of South.
To solve this problem, we can use the Pythagorean theorem and trigonometry to find the distance and direction of the trip, relative to the south.
1. Draw a diagram: Start by drawing a diagram that represents the boat's path. Label the starting point as A, the destination point as B, and the direction south as S.
2. Calculate the distance: The distance can be found using the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In this case, the eastward distance is 19 km, and the southward distance is 32 km. Using the Pythagorean theorem, we can calculate the distance as follows:
distance^2 = (eastward distance)^2 + (southward distance)^2
distance^2 = 19^2 + 32^2
distance^2 = 361 + 1024
distance^2 = 1385
Therefore, the distance is the square root of 1385:
distance = sqrt(1385) ≈ 37.23 km
So, the distance of the trip, relative to the south, is approximately 37.23 km.
3. Calculate the direction: To find the direction, we can use trigonometry. The angle between the south direction and the hypotenuse of the right triangle formed by the boat's path is the desired direction.
We can use the tangent function to find this angle, which is defined as the ratio of the opposite side (eastward distance) to the adjacent side (southward distance). Therefore, the angle (θ) can be calculated as follows:
tan(θ) = (eastward distance) / (southward distance)
tan(θ) = 19 / 32
θ = tan^(-1)(19/32)
Using a calculator or trigonometric tables, we can find that θ is approximately 30.96 degrees.
So, the direction of the trip, relative to the south, is approximately 30.96 degrees east of south.
To summarize:
- The distance of the trip, relative to the south, is approximately 37.23 km.
- The direction of the trip, relative to the south, is approximately 30.96 degrees east of south.