Joe walks 5 miles in the direction degrees North of East, and then 10 miles in the direction 20 degrees West of North.
(a) How far is Joe from his starting position?
(b) In what direction is Joe from his starting position?
You left out an important number that is needed to compute the answer. It should follow the first word "direction". How many degrees is the first direction North of East?
Hi, sorry it was 30 degrees North of East.
total displacement north
= 5 sin 30 + 10 cos20
= 2.5 + 9.397
= 11.897 miles
total displacement east
= 5 cos30 - 10 sin20
= 4.33 - 3.42
= 0.91 miles
(a) sqrt[(11.897)^2 + (0.91)^2]
= 11.93 miles
(b) direction = tan^-1 0.91/11.897
E of N
= 4.37 deg E of N
Hi, thank you so much. Everything I tried hit all around the borders, but just was not working out correctly. I really appreciate your help.
To find the answer to these questions, we can visualize Joe's movements using vectors and trigonometry.
(a) To find how far Joe is from his starting position, we can break down his movements into two components: north-south and east-west. Let's calculate the north-south component first:
Joe walks 5 miles in the direction degrees North of East. This gives us the east-west component which can be calculated using trigonometry. The angle between the east direction and Joe's 5-mile displacement is (90 degrees - angle degrees North of East).
Using the trigonometric function sine (sin), we can find the east-west component:
east-west component = 5 miles * sin(angle degrees North of East)
Next, Joe walks 10 miles in the direction 20 degrees West of North. This gives us the north-south component which can be calculated using trigonometry. The angle between the north direction and Joe's 10-mile displacement is (90 degrees - angle degrees West of North).
Using the trigonometric function cosine (cos), we can find the north-south component:
north-south component = 10 miles * cos(angle degrees West of North)
Now, to find the total displacement, we can use Pythagoras' theorem, which states that the square of the hypotenuse (total displacement) is equal to the sum of the squares of the two sides (north-south and east-west components). So, the formula to find the total displacement is:
Total displacement = square root of ((east-west component)^2 + (north-south component)^2)
We can substitute the values we calculated to find the final answer.
(b) To determine the direction Joe is from his starting position, we can use the inverse tangent function (arctan) to find the angle of the displacement vector with respect to the east direction.
Angle direction = arctan(north-south component / east-west component)
Now, we will substitute the values we calculated to find the final answer.
Let's do the calculations:
For the north-south component:
angle degrees West of North = 90 degrees - 20 degrees = 70 degrees
north-south component = 10 miles * cos(70 degrees)
For the east-west component:
angle degrees North of East = 90 degrees - 70 degrees = 20 degrees
east-west component = 5 miles * sin(20 degrees)
To find the total displacement:
Total displacement = square root of ((east-west component)^2 + (north-south component)^2)
To find the angle direction:
Angle direction = arctan(north-south component / east-west component)