Question

Suppose a rectangular plot is to be fenced in, with the fencing on each of the four sides (labeled east, west, north, south) costing as follows: north: $4 per foot south: $5 per foot east: $3 per foot west: $5 per foot and the total cost of fencing is to be $700. What is the maximum area in ft2 of such a plot?

Answer 1

A = 1701,38 ft²

Dimensions :

x (north and south sides ) = 38.89 ft

y ( east and west sides ) = 43,75 ft

Explanation:

North and south (sides of same length) equal "y" cost (4 + 5 ) = 4,5 $/ft²

East and west (sides of same length) equal "x" cost ( 3 + 5 ) = 4 $ /ft²

Equation of cost is

C = Cost of (north + south ) + Cost (east + west)

C = 2 * 4,5 * x + 4*2* y

C = 9x + 8y

700 = 9x + 8y ⇒ y = ( 700- 9x)/ 8

A = x*y

A(x) = x * ( 700 - 9x ) /8

A(x) = ( 700 x -9x²) / 8 A´(x) = ( 700 - 18 x )/ 8 A´(x) = 0

( 700 - 18 x )/ 8 = 0 ⇒ 700 - 18 x = 0 ⇒ x = 700/18

x = 38.89 ft

y = ( 700 - 9x )/8 ⇒ y = 349.99 / 8 ⇒ y = 43.75

And maximum ara is

A = x*y A = 38.89 * 43.75 = 1701,38 ft²