You are camping with two friends, Joe and Karl. Since all three of you like your privacy, you don't pitch your tents close together. Joe's tent is 19.5 from yours, in the direction 21.5 north of east. Karl's tent is 41.0 from yours, in the direction 41.0 south of east.
What is the distance between Karl's tent and Joe's tent?
The three tents form a triangle with side lengths a = 19.5, b = 41.0, and included angle between them of 62.5 degrees. You want the length of the thrid side, c.
The angle between the directions to the two tents, as seen from "my" tent, is 62.5 degrees. For the distance between them (c), use the law of cosines
c^2 = a^2 + b^2 - 2 a b cos62.5
= 19.5^2 + 41^2 - 2*19.5*41*(0.4617)
= 1323.0
c = 36.37 m
thanks you so much
To find the distance between Karl's tent and Joe's tent, we can use the concept of relative displacement.
First, let's represent the given information graphically. Draw a diagram with your tent as the reference point:
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Joe => |---------> 19.5 m @ 21.5° N of E
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You => | Karl => 41.0 m @ 41.0° S of E
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Now, let's break down the given information into its vector components:
For Joe's tent:
- Magnitude (distance from your tent): 19.5 m
- Direction: 21.5° north of east
For Karl's tent:
- Magnitude (distance from your tent): 41.0 m
- Direction: 41.0° south of east
Next, we need to convert the direction given in terms of cardinal directions (north, south, east, west) into standard mathematical directions (measured counterclockwise from the positive x-axis).
For Joe's tent:
- Direction: 21.5° north of east
- Convert it to standard mathematical direction:
- Subtract 90° since north is 90° counterclockwise from the positive x-axis.
- So, the standard mathematical direction is 90° - 21.5° = 68.5° counterclockwise from the positive x-axis.
For Karl's tent:
- Direction: 41.0° south of east
- Convert it to standard mathematical direction:
- Add 90° since south is 90° clockwise from the positive x-axis.
- So, the standard mathematical direction is 90° + 41.0° = 131.0° counterclockwise from the positive x-axis.
Now, we have the magnitude and direction of both vectors. To find the distance between Karl's tent and Joe's tent, we can use the law of cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where:
- c is the distance between Karl's tent and Joe's tent.
- a is the magnitude of Joe's tent's vector.
- b is the magnitude of Karl's tent's vector.
- C is the angle between the vectors (measured counterclockwise from Joe's tent to Karl's tent).
In this case, a = 19.5 m, b = 41.0 m, and C = (131.0° - 68.5°) = 62.5° (measured counterclockwise from Joe's tent to Karl's tent).
Now, we can plug in these values into the equation:
c^2 = (19.5^2) + (41.0^2) - 2(19.5)(41.0) * cos(62.5°)
Solving this equation will give us the squared distance between Karl's tent and Joe's tent. Taking the square root of this value will give us the actual distance.