Amazon is experimenting with drone delivery and you are asked to determine a drone's delivery route.

Question

Amazon is experimenting with drone delivery and you are asked to determine a drone’s delivery route. The drone will leave the Amazon warehouse, make two deliveries, and then return to the warehouse. The first delivery is 5.42 miles from the warehouse in a direction 35.0^O south of the west. The second delivery is 7.19 miles, in a direction 22.0^O south of east, from the location of the first delivery. After the drone makes its second delivery, find: 

  (a) how far, in miles and

  (b) in what direction, 

the drone must fly to return to the warehouse.

Answer 1

Dear Maleeha,

to solve this problem, we can separate the motion of the drone into the x and y axis aligned with the North and East respectively.

On the first trip, the drone moves a total distance of 5.42 miles with an angle of 35 degrees while the second trip took 7.19 miles with an angle of 22 degrees. To find its x and y coordinates we use trigonometry, the distance along the x direction is 5.42 cos(35) + 7.19 cos(22) = 11.1063. and along the y direction is -5.42 sin(35) - 7.19 sin(22) = -5.80221. Note that the y direction is negative since we move southwest.

The total distance and angle to return are given by

D = Sqrt(11.1063²+(-5.80221)²) = 12.5306 miles

Theta = ArcTan(-5.80221/11.1063) = -27.5837 + 180 = 152.416 degrees

Note that we added 180 degrees to find the return direction to the starting point.

If you have any more questions please sent me a message and I would be happy to help.

Answer 2

This looks like a fun take on a classic vector addition problem. You can treat each displacement like a vector and add them to get the total displacement. If the drone flies straight back to the warehouse it should travel magnitude of the total displacement. I'm going to treat East as the positive x-axis.

Δx₁ = <5.42 mi ⋅ cos(215°), 5.42 mi ⋅ sin(215°) >,

Δx₂ = <7.19 mi ⋅ cos(338°), 7.19 mi ⋅ sin(338°) >,

a) d = |Δx₁ + Δx₂| ≈ 6.21 mi

b) With this one I would say draw a vector diagram to get the direction correct.

Δx₁ + Δx₂ ≈ <2.23 mi, -5.80 mi>

This looks like it would be in Quadrant VI, or arctan(2.23/5.80) ≈ 21.03° East of South. The drone needs to fly in the opposite direction so I would say 21.03° West of North.

Please double check this with a vector diagram to make sure everything is right.