SOLUTION: A trough for feeding cattle is 16 feet long and it's cross sections are isosceles triangles with two equal sides being 18 inches. The angle between the two equal sides is theta.

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Question 1060565: A trough for feeding cattle is 16 feet long and it's cross sections are isosceles triangles with two equal sides being 18 inches. The angle between the two equal sides is theta.
A) express the volume of the trough as a function of theta/2
B) express the volume of the trough as a function of theta and determine the value of theta so the volume is maximum.
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
If the isosceles triangle is cut in half such that +theta+ is bisected, then
the side adjacent to +theta%2F2+ is +18%2Acos%28theta%2F2%29+ and the side opposite
theta%2F2 is +18%2Asin%28theta%2F2%29+:
+Area_half+=+%281%2F2%29%2A%2818%2Asin%28theta%2F2%29%29%2A%2818%2Acos%28theta%2F2%29%29+
--
The full area of the isosceles triangle is twice this:
+Area+=+18%2Asin%28theta%2F2%29+%2A+18%2Acos%28theta%2F2%29+
+Area+=+324%2Asin%28theta%2F2%29%2Acos%28theta%2F2%29+ (this is sq in)
Adjust the formula to give us sq ft (sq ft = sq in / 144):
+Area+=+%28324%2F144%29%2Asin%28theta%2F2%29%2Acos%28theta%2F2%29+ sq ft
—
The trough volume is just 16 ft times this area:
+Volume+=+16%2A%28324%2F144%29%2Asin%28theta%2F2%29%2Acos%28theta%2F2%29+ +ft%5E3+
—
Ans:
+Volume+=+36%2Asin%28theta%2F2%29%2Acos%28theta%2F2%29+ +ft%5E3+
—
To get the volume as a function of +theta+ apply the half-angle formula:
+sin%28alpha%29+=+2%2Asin%28alpha%2F2%29%2Acos%28alpha%2F2%29+
—
Ans:
+Volume+=+18%2Asin%28theta%29+ +ft%5E3+ +ft%5E3+
The maximum volume will be at +theta=90+degrees where sin( ) has its max value of 1, which gives a volume of 18 +ft%5E3