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 16.0 from yours, in the direction 22.0 north of east. Karl's tent is 39.5 from yours, in the direction 39.5 south of east.
What is the distance between Karl's tent and Joe's tent?
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Didn't you ask a very similar problem yesterday? Only the numbers have changed. See many of the Related Questions below.
You have two sides and the included angle of a triangle. The two sides are a = 12 and b = 39.5 long, and the included angle is C = 61.5 degrees. Use the law of cosines for the third side length, c.
c^2 = a^2 + b^2 -2ab cos C
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put your finger in the dogs butt
To find the distance between Karl's tent and Joe's tent, we can use the concept of vector addition. First, we need to convert the given information into vector form.
Let's assume that your tent represents the origin (0,0).
Joe's tent is 16.0 units away in the direction 22.0 degrees north of east. We can represent this as a vector using the magnitude (16.0) and the direction (22.0°).
To convert the direction into coordinates, we need to split it into its x and y components. The x-component represents the east direction, and the y-component represents the north direction.
To find the x-component, we use the cosine function:
cos(22.0°) = adjacent/hypotenuse = x/16.0
x = 16.0 * cos(22.0°) ≈ 14.769
To find the y-component, we use the sine function:
sin(22.0°) = opposite/hypotenuse = y/16.0
y = 16.0 * sin(22.0°) ≈ 5.828
So, Joe's tent is located at approximately (14.769, 5.828).
Similarly, Karl's tent is 39.5 units away in the direction 39.5 degrees south of east. We can calculate the x and y components for Karl's tent in the same manner.
To find the x-component for Karl's tent, we use the cosine function:
cos(39.5°) = adjacent/hypotenuse = x/39.5
x = 39.5 * cos(39.5°) ≈ 30.058
To find the y-component for Karl's tent, we use the sine function:
sin(39.5°) = opposite/hypotenuse = y/39.5
y = 39.5 * sin(39.5°) ≈ -25.265
So, Karl's tent is located at approximately (30.058, -25.265).
Now, we can calculate the distance between Joe's and Karl's tents using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((30.058 - 14.769)^2 + (-25.265 - 5.828)^2)
Distance ≈ sqrt((15.289)^2 + (-31.093)^2)
Distance ≈ sqrt(234.960 + 966.752)
Distance ≈ sqrt(1201.712)
Distance ≈ 34.65
Therefore, the distance between Karl's tent and Joe's tent is approximately 34.65 units.