Answer :
Answer:
The cost of the least expensive fence is $400
Step-by-step explanation:
Let the side of the area facing north and south = x
Let the side of the area facing west and east = y.
- Area to be enclosed [tex]=5000$ ft^2[/tex]
- Therefore: Area=xy=5000
Perimeter, P(x,y)=2(x+y)=2x+2y
Fencing material costs $1 per foot for the two sides facing north and south and $2 per foot for the other two sides.
Therefore, Cost of Fencing=($1 X 2x)+($2 X 2y)=
C(x,y)=2x+4y
We can write the cost function as a function of one variable by substituting for y.
Recall: xy=5000
[tex]y=\dfrac{5000}{x}[/tex]
Therefore:
[tex]C(x)=2x+4(\dfrac{5000}{x} )\\C(x)=\dfrac{2x^2+20000}{x}[/tex]
To determine the cost of the least expensive, we minimize C(x) by taking its derivative and solving for its critical points.
[tex]C'(x)=\dfrac{2x^2-20000}{x^2}\\$When C'(x)=0\\2x^2-20000=0\\2x^2=20000\\x^2=10000\\x^2=100^2\\x=100$ ft[/tex]
Therefore, the cost of the least expensive fence will be:
[tex]C(100)=\dfrac{2(100)^2+20000}{100}\\=\$400[/tex]