Question

A landscape architect plans to enclose a 2600 square foot rectangular region in a botanical garden. She will use shrubs costing $25 per foot along three sides and fencing costing $10 per foot along the north side.

Find the minimum total cost.

Answer 1

$4,265.90

Explanation:

-Let X and Y be the two sides of the rectangle and C be the cost function.

-The cost function can then be expressed as:

C=25(2y+x)+10x

C=50y+35x ...(i)

#The region's area is equated as:

xy=2600 ...(ii)

#We make Y the subject of the formula in i and substitute in ii;

y=2600/x

=>C=50y+35x

C=50(2600/x)+35x

C=130000/x+35x

#We get the first derivative to determine the critical points:

dC/dx=-130000/x^2+35

#Set dC/dx=0

130000/x^2=35

x=60.94 ft

y=2600/x=2600/60.94=42.66 ft

-The minumum is therefore calculated as:

C=50y+35x

=50(42.66)+35(60.94)

=$4,265.90

Hence, the minimum cost is $4,265.90