Ajax is 8 km due west of Oshawa. Uxbridge is 16 km NW of Oshawa. How far is it from Ajax to Uxbridge? Explain whether you have enough information to solve this problem.
my answer was 13.856 km but not confident how I found the angle oshawa to begin with. I used Cos Oshawa = a/h = 8/16, cos-1 0.5=60 degrees.
Please confirm if done correctly...many thanks.
angle AOU = 45 degrees
ao = 8
ou = 16
law of cosines
au^2 = ao^2 + ou^2 - 2*ao*ou*cos 45
au^2 = 64 + 256 - 2*8*16 * .707
au^2 = 139
au = 11.8
Hi - How did you get 45 degrees? Think that's the part I'm lost with.
The angle between west and northwest is 45 degrees, halfway from west to north
Yes but why is it 45 degrees? I'm dividing 8 into 16 = .5 then cos-1 .5 = 60. Nothing I do is giving me 45 degrees. Hope you get notified when a thread is updated, I'm inclined to repost this incase.
Thanks again for your great answering.
Label and sketch the triangle as I did. You will see that the angle between ao and ou is 45 degrees
To solve this problem, you can use the concept of vector addition. Let's break down the information given:
1. Ajax is 8 km due west of Oshawa.
2. Uxbridge is 16 km NW of Oshawa.
First, let's find the distance and direction from Oshawa to Uxbridge. Since Uxbridge is northwest of Oshawa, the angle between the direction of Oshawa and the direction of Uxbridge is 45 degrees (since northwest is halfway between north and west).
Using basic trigonometry, we can determine that the horizontal component of Uxbridge's displacement is given by: cos(45 degrees) * 16 km = 11.315 km
Next, let's find the distance and direction from Oshawa to Ajax. Since Ajax is due west of Oshawa, the angle between them is 0 degrees.
By applying the same trigonometric logic as before, we can determine that the horizontal component of Ajax's displacement is given by: cos(0 degrees) * 8 km = 8 km
Now we can use vector addition to find the distance and direction from Ajax to Uxbridge. We have the horizontal components for both displacements: 8 km (from Oshawa to Ajax) and 11.315 km (from Oshawa to Uxbridge).
To find the distance between Ajax and Uxbridge, we can subtract the horizontal component of Ajax's displacement from the horizontal component of Uxbridge's displacement:
Distance = 11.315 km - 8 km = 3.315 km
Therefore, the distance between Ajax and Uxbridge is approximately 3.315 km.
In summary, you have correctly calculated the angle of Oshawa as 60 degrees using the cosine function. However, since Uxbridge is northwest of Oshawa, the angle should be 45 degrees. Using the correct angle, you can apply trigonometry to find the horizontal component of the displacement and then subtract to get the distance between Ajax and Uxbridge.