Farmer Steve plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180000 square meters in order to prove enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?

Question

Farmer Steve plans to fence a rectangular pasture adjacent to a river.  The pasture must contain 180000 square meters in order to prove enough grass for the herd.  What dimensions would require the least amount of fencing if no fencing is needed along the river?

anonymous · Answer

This is another optimization problem, there were a few similar ones this morning.
180,000 = 2x + y, where y = length and x = width. The second length is bordered by the river, so it doesn't take fencing.
A = x * y
y = 180,000 - 2x
A = (180,000-2x)(x)
A = 180,000x -2x^2
Differentiate
A' = 180,000 - 4x
Set A' = 0
0 = 180,000 - 4x
4x = 180,000
x = 180,000/4 = 45,000
y = 180,000 - 2x = 180,000 - 90,000
y = 90,000
x = 45,000

anonymous · Answer

I am not good at all with optimization promblems. When the material is taught it seems like it should come very easy but doesn't. Practice makes perfect! Thank you so much for all of your help.

anonymous · Answer

You're welcome.
Optimization problems have several different kinds that all work the same way mathematically but require insight into setting them up. I would suggest looking in your book's exercises for the various setups and try solving them independently before the test.