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

Answer:

A = 1701,38 ft²

Dimensions :

x (north and south sides )  =  38.89 ft

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

Step-by-step 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