\
The x and y component of a certain force are measured and found to be 68 (+/-3) N and 42 (+/-) N, respectively. Calculate the direction and magnitude of this force, expressing your results to the proper number of significant figures and showing the error in both magnitude and angle. Note that the error in the latter is obtained from the error in its tangent. Devise a procedure for doing this and show your work.
\
Step 1:
\
The x and y components of the force are ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=7d8b7e20be5abf8b363ad3df3d5834ab.png\")
and ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=f6133ad7cdb3f27a7f43d2dd35d0a37d.png\")
.
\
Find the magnitude and direction of the resultant force.
\
When ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=37ca7eaedcca6a5a6446929dee61cf83.png\")
and ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=a893cbd5393ce55663235d76a3a94f91.png\")
, then the magnitude of the resultant force is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=14133669f2a783f1d1d911b7f294abc6.png\")
.
\
![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=0f1ed622e8c1853f041e0ac566487bb4.png\")
\
The magnitude of the resultant force is 79.92 N.
\
When ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=37ca7eaedcca6a5a6446929dee61cf83.png\")
and ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=a893cbd5393ce55663235d76a3a94f91.png\")
, then the direction of the resultant force with respect to the forizontal direction is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=4715c71bb5f2e80c82b11d2c1793e8df.png\")
.
\
![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=9334d3dbe46717b67f39390b55add371.png\")
\
The direction of the resultant force is 31.70°.
\
The resultant force is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=fa4bc53936dafdcb5226c9db75dfeea5.png\")
.
\
Step 2:
\
The maximum angle occurs when x is a minimum and y is a maximum.
\
Minimum value of x is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=0ab9d259de4b6fb451d85b7fb439900f.png\")
N.
\
Maximum value of y is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=d1c7a98f1bc600d658f5917f39bd4b5b.png\")
N.
\
The magnitude of the resultant force is ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=14133669f2a783f1d1d911b7f294abc6.png\")
.
\
![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=72fe116bb35719e750bce7b272c7a74c.png\")
\
The magnitude of the resultant force is 78.49 N.
\
The direction of the resultant force is ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=4715c71bb5f2e80c82b11d2c1793e8df.png\")
.
\
![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=c0a2d1ce7f2a5428b8f173a191833a46.png\")
\
The direction of the resultant force is 34.095°.
\
The maximum resultant force is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=b4afe290265cfbfc095935001d47121b.png\")
.
\
Step 3:
\
The minimum angle occurs when x is a maximum and y is a minimum.
\
Maximum value of x is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=18affe6b8aa1268d0f818587fdae71c7.png\")
N.
\
Minimum value of y is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=10e23384cacf0eb152980f0e48ffcbab.png\")
N.
\
The magnitude of the resultant force is .
\
![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=0f88ab0793053d8eb6cb3830ca6cd554.png\")
\
The magnitude of the resultant force is 81.49 N.
\
The direction of the resultant force is .
\
![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=f43f4c7884557b065b15cb48fe7feb73.png\")
\
The direction of the resultant force is 29.396°.
\
The minimum resultant force is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=c38989a7d49e681ded013e2e2d4c272b.png\")
.
\
Possible error in the magnitude of the resultant force is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=1e707369a4b519861612a10987399289.png\")
.
\
Possible error in the direction of the resultant force is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=e601c97fe3e13fc1d73fced687d9b992.png\")
.
\
Therefore the resultant force with possible error is ![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=51d6e1e0f148c58ebdacbccb01920be5.png\")
.
\
Solution:
\
The resultant force is ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=fa4bc53936dafdcb5226c9db75dfeea5.png\")
.
\
The maximum resultant force is ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=b4afe290265cfbfc095935001d47121b.png\")
.
\
The minimum resultant force is ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=c38989a7d49e681ded013e2e2d4c272b.png\")
.
\
The resultant force with possible error is ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=51d6e1e0f148c58ebdacbccb01920be5.png\")
.