Question

A landscape architect wishes to enclose a rectangular garden on one side by a brick wall costing $21 per foot and on the other three sides by a metal fence costing $7 per foot. If the area of the garden is 800 ft, find the dimensions of the garden minimizing the cost. (Let x be the length of the brick wall and y be the length of an adjacent side in feet.)

Answer 1

here are two equations that can be figured out from the question
21*x + 7*y = total cost
x*y = 800

can you figure out how to relate them into one equation, in order to solve for the minimum cost?

Answer 2

length of the brick wall =x
length of an adjacent side =y
area=x*y=800

x*y=800
brick wall cost=$21*x
metal fence cost=$7*y+$7y+*7x
total cost=$28*x+$14*y
x*y=800
y=800/x
total cost=$28*x+$14*800/x

total cost=28x+11200/x
differntiate
d(total cost)/dx =28-11200/x^2 =(28*x^2-11200)/x^2
tofind value of x put equation =0
(28*x^2-11200)/x^2=0
(28*x^2-11200)=0
x^2=11200/28=400
x=20
now find
d^2(total cost)/dx^2 =-11200* (-2)/x^3
now calculate its value at x=20
=22400/20^3=22400/8000 >0
if second derivative is > 0 then function will have minimum value at x=20

x=20feet
x*y=800
y=800/20=40feet
x= 20 feet
y=40 feet

Answer 3

@marjorie