How to solve this questions? "A fence is to be built to enclose a rectangular area of 800 square meter. The fence along three sides is to be made of material that costs 6 dollars per meter, and the material for the fourth side costs 18 dollars per meter. Find the dimensions of the rectangular that will allow the most economical fence to be built. What is the most economical dimension if the cost of all four side of the fence is the same? "

Question

sooobored · Answer

First, we start by defining some variables
let x be the width of our rectangle
let y be the length of our rectangle 
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sooobored · Answer

Let "C" be the total cost of the fencing material
forgot to include, variables x and y are in meters

Then, we look at the given information and attempt to place them into the form of an equation 
"A fence is to be built to enclose a rectangular area of 800 square meter" means 
800= x*y
since the area of a rectangle is given by A= w*l

"The fence along three sides is to be made of material that costs 6 dollars per meter, and the material for the fourth side costs 18 dollars "

this means that  2x + 1y side are 6 dollars per meter and the remaining side y is 18 dollars per meter.
we can represent this as 6(2x+y) is equal to the total cost of the three sides and 18y is the total cost of the 4th side

and putting them together to get the total cost of the fencing material "C"
C= 6(2x+y)+18y

sooobored · Answer

So now we look at what the question is asking us. 
"What is the most economical dimension if the cost of all four side of the fence is the same? "

in other words, what are the dimensions of the rectangle in order to get the lowest cost C

sooobored · Answer

but wait, 
C is a "function" of two variables x and y

we cant do a single derivative in order to determine a critical point--- well you could, but i dont think it would help much since there is a relationship between x and y

sooobored · Answer

Thankfully, we remember that the area of the rectangle is also a function of x an y

this means we can solve for either x or y  and then substitute that into the main cost "C" equation 

so x=800/y or y=800/x
hooray

sooobored · Answer

now, since this is an optimization problem where we're looking for the most economical cost, lets assume that the graph of the length of |dw:1477986514991:dw|one side vs total cost will look something like this