SOLUTION: PLEASSSSEE HELP . Determine the dimensions of a rectangle of area 60 in^2 that is inscribed in a circle or radius 6.5 in.

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Question 850285: PLEASSSSEE HELP . Determine the dimensions of a rectangle of area 60 in^2 that is inscribed in a circle or radius 6.5 in.
Found 2 solutions by josgarithmetic, josmiceli:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Just solved it nearly to completion except for some simplifications for one of the dimensions.

The same question answered. Number 850276

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +x+ = one of the dimensions of the rectangle
+60%2Fx+ = the other dimension
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The center of the circle that this rectangle is
inscribed in is located where the diagonals cross
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The length of a diagonal is:
+d+=+sqrt%28+%28+60%2Fx+%29%5E2+%2B+x%5E2+%29+
+d+=+sqrt%28+3600%2Fx%5E2+%2B+x%5E2+%29+
---------------------------
This diagonal is the same as the diameter of the
circle, which is
+2r+=+2%2A6.5+
+2r+=+13+
Now I can say:
+13+=+sqrt%28+3600%2Fx%5E2+%2B+x%5E2+%29+
Square both sides
+169+=+3600%2Fx%5E2+%2B+x%5E2+
Let +x%5E2+=+z+
+169+=+3600%2Fz+%2B+z+
Multiply both sides by +z+
+169z+=+3600+%2B+z%5E2+
+z%5E2+-+169z+%2B+3600+
----------------------
+z+=+%28-b+%2B-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+
+a+=+1+
+b+=+-169+
+c+=+3600+
+z+=+%28+169+%2B-+sqrt%28+%28-169%29%5E2+-+4%2A1%2A3600+%29%29+%2F+%282%2A1%29+
+z+=+%28+169+%2B-+sqrt%28+28561+-+14400+%29%29+%2F+2+
+z+=+%28+169+%2B-+sqrt%28+14161+%29%29+%2F+2+
+z+=+%28+169+-+119%29+%2F+2+
+z+=++50%2F+2+
+z+=+25+
and
+z+=+x%5E2+
+x+=+sqrt%2825%29+
+x+=+5+
+60%2Fx+=+60%2F5+
+60%2Fx+=+12+
The dimension are:
5 x 12 in2
check:
+5%2A12+=+60+
OK
Note that is you go back and pick the
positive +sqrt%28+14161+%29+, then you
get the same answers