A ship sails 15.0 mi on a course S40°10’W and then 21.0 mi on a course N28°20’W. Find the distance and direction of the last position from the first.
convert to (x,y) values and add:
15@S40°10'W = (-9.675,-11.463)
21@N28°20'W = (-9.967,18.484)
sum = (-19.642,7.021)
or,
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To find the distance and direction from the last position to the first position, we can use vector addition.
First, we need to convert the given courses to Cartesian coordinates.
For the first course, S40°10'W, we can break this down into two components:
- The southern component: 15.0 miles * sin(40°10')
- The western component: 15.0 miles * cos(40°10')
Using a calculator, we find that the southern component is approximately 9.664 miles, and the western component is approximately 11.446 miles.
For the second course, N28°20'W, we can also break this down into two components:
- The northern component: 21.0 miles * sin(28°20')
- The western component: 21.0 miles * cos(28°20')
Again, using a calculator, we find that the northern component is approximately 9.884 miles, and the western component is approximately 18.496 miles.
Now, let's add up the components. The total southern component is 9.664 miles, and the total northern component is 9.884 miles. Adding them together, we get a net southern component of 9.664 miles - 9.884 miles = -0.22 miles.
Similarly, the total western component is 11.446 miles + 18.496 miles = 29.942 miles.
To find the distance and direction from the last position to the first position, we can use the Pythagorean theorem and trigonometry. The distance, d, is given by:
d = sqrt(southern_component^2 + western_component^2)
Using the values we calculated, the distance is:
d = sqrt((-0.22 miles)^2 + (29.942 miles)^2)
≈ sqrt(0.0484 miles^2 + 896.329364 miles^2)
≈ sqrt(896.377764 miles^2)
≈ 29.93 miles
Therefore, the distance from the last position to the first position is approximately 29.93 miles.
To find the direction, we can use the trigonometric function arctan. The direction, θ, is given by:
θ = arctan(western_component / southern_component)
Using the values we calculated, the direction is:
θ = arctan(29.942 miles / -0.22 miles)
≈ arctan(-135.645454545)
≈ -89.9304°
Therefore, the direction from the last position to the first position is approximately 89.9304° west of south or S89°55'W.