Question

A rectangular field is to be enclosed on four sides with a fence. Fencing costs $4 per foot for two opposite sides, and $7 per foot for the other two sides. Find the dimensions of the field of area 740 ft 2 that would be the cheapest to enclose.

A. 36 ft @ $4 by 20.6 ft @ $7

B. 20.6 ft @ $4 by 36 ft @ $7

C. 47.6 ft @ $4 by 15.5 ft @ $7

D. 15.5 ft @ $4 by 47.6 ft @ $7

Answer 1

Length=X

Width=y

Area of rectangle=$740ft^2$

$XY=740.....(1)$

Length of rectangle is 4per foot and width is 7per foot

$C(x,y)=4x+7y\\=4x+7x(\frac{740}{X})\\=4x+\frac{5180}{x}$

$xy=740\\y=\frac{740}{x}$

For the cheapest value

$c(x)=0\\\frac{d}{dx}(4x+\frac{5180}{x})=0\\ \frac{5180}{x^2}=4$

$x^2=\frac{5180}{4}\\ X=36feet$

$xy=740\\y=\frac{740}{35.906}=20.6feet$

Answer is

A. 36 ft @ $4 by 20.6 ft @ $7