Hi - I'm reposting this thread please read the bottom...thanks.
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.
math - Damon, Sunday, August 21, 2011 at 2:44pm
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
math - Dee, Sunday, August 21, 2011 at 3:14pm
Hi - How did you get 45 degrees? Think that's the part I'm lost with.
math - Damon, Sunday, August 21, 2011 at 4:16pm
The angle between west and northwest is 45 degrees, halfway from west to north
math - Sunday, August 21, 2011 at 4:51pm
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.
draw a sketch and you will see that the angle between ao and ou or angle AOU is that between west and northwest or 45 degrees
Please look up "law of cosines" with Google.
Like try this calculator. You have SAS (side, angle, side)
how will I see that the angle is 45 degrees? Don't we have to calculate that? This is what I'm asking you about. How is <O calculated so it comes up to 45 degrees?
I have 2 sides and have to find the angle as it's not given before I can proceed to find the third side. I'm asking how you got 45 so I can understand where I went wrong. Thanks for recommending google...that was very helpful in providing clarity to this particular question.
one side goes west
the second side goes north west, 45 degrees clockwise from the first side.
There is no calculation of 45 degrees. It is given by saying west and northwest
Ahhh...I see! Thank you for helping, I would have never have worked this out correctly.
To find the distance from Ajax to Uxbridge, we can use the law of cosines.
First, let's draw a diagram to visualize the problem. We have a triangle with sides AO, OU, and AU. Oshawa is located at point O, Ajax is at point A, and Uxbridge is at point U. AO is 8 km (west), OU is 16 km (northwest), and we want to find AU.
To use the law of cosines, we need to know the angle between sides AO and OU. In the initial answer, the angle was assumed to be 45 degrees. However, the angle between west and northwest is actually 135 degrees, which can be found by subtracting 90 degrees (north) from 225 degrees (southwest).
Using the law of cosines, we have:
AU^2 = AO^2 + OU^2 - 2 * AO * OU * cos(angle AOU)
Substituting the given values, we get:
AU^2 = 8^2 + 16^2 - 2 * 8 * 16 * cos(135)
Evaluating the expression, we have:
AU^2 = 64 + 256 - 256 * cos(135)
Next, we need to calculate the cosine of 135 degrees. Since the cosine function expects radians as an input, we first convert 135 degrees to radians (2π radians = 360 degrees):
135 degrees * (2π radians / 360 degrees) = 3π/4 radians
cos(135 degrees) = cos(3π/4 radians) = -cos(π/4 radians) = -1/√2
Substituting this value in, we have:
AU^2 = 64 + 256 - 256 * (-1/√2)
Simplifying, we get:
AU^2 = 139
Finally, we take the square root of both sides to find AU:
AU ≈ √139 ≈ 11.8 km
Therefore, the distance from Ajax to Uxbridge is approximately 11.8 km.