Answer:
150 feet by 300 feet.
Step-by-step explanation:
The fence is to enclose a rectangular area of 45,000 ft squared.
If the dimensions of the rectangle are x and y
Area of a rectangle = xy
Perimeter of the Rectangle =2x+2y
Fencing material costs $ 3 per foot for the two sides facing north and south and $6 per foot for the other two sides.
Substitute [tex]x=\frac{45000}{y}[/tex] into the Cost to get C(y)
C=12x+6y
[tex]C(y)=12(\frac{45000}{y})+6y\\C(y)=\frac{540000+6y^2}{y}[/tex]
The value at which the cost is least expensive is at the minimum point of C(y), when the derivative is zero.
[tex]C^{'}(y)=\dfrac{6y^2-540000}{y^2}[/tex]
[tex]\dfrac{6y^2-540000}{y^2}=0\\6y^2-540000=0\\6y^2=540000\\y^2=\frac{540000}{6} =90000\\y=\sqrt{90000}=300[/tex]
Recall,
[tex]x=\frac{45000}{y}=\frac{45000}{300}=150[/tex]
Since x=150, y=300
The dimensions that will be least expensive to build is 150 feet by 300 feet.
