Optimization:A rose garden and a lawn

[wind]

Hi, I am having trouble with this question can someone help? Thanks

A landscape architect is creating a rectangular rose garden to be located in a local park. The rose garden is to have an area of 60m² and be surrounded by a lawn. The surrounding lawn is to be 10m wide on the north and south side of the garden and 3m wide on the east and west sides. find the dimensions of the rose garden if the total area of the garden and lawn together is to be a minimum.

when they say "10m wide on the north and south " they mean 10m from the garden on both sides, or the length is 10?

Let L rep the length of the garden
Lat w rep the width of the garden

Garden

A=Lw
60=Lw
60/w=L

Total Area

A=(20+L)(6+w)
A=(20+(60/w))(6+w)
A=[(20w+60)/w](6+w)
A=(120w+20w²+360+60w)/w
A=(180w+20w²+360)/w

A'=[(180+40w)(w) - (180w+20w²+360)(1)]/w²
A'=[180w+40w²-180w-20w²-360]/w²
A'=(20w²-360)/w²

0=(20w²-360)/w²
0=20w²-360
360=20w²
18=w²
root18=w

 

[skeeter]

let w = width of garden (let that run north/south)
then 60/w = length of garden (let that run east/west)

let A = area of the garden + lawn

A = (w + 20)(60/w + 6)

A = 6w + 1800/w + 180

now ... find dA/dw and minimize.

 

[wind]

A = (w + 20)(60/w + 6)
A = 6w + 1800/w + 180

Thanks skeeter but why is is 1800/w, is'nt it 1200/w, 20*(60/w)?

 

[skeeter]

my mistake ... I was thinking 30x60 instead of 20x60

 

[wind]

A = (w + 20)(60/w + 6)
A = 60w + 1200/w + 180
A = 60w + 1200w^-1 + 180
A' = 60 - 1200/w²
0= 60 - 1200/w²
-60=- 1200/w²
-60=- 1200/w²
-60w²=-1200
w²=20
w=10root2

I think I got it thanks skeeter

 

[skeeter]

A = (w + 20)(60/w + 6)
A = 60w + 1200/w + 180
A = 60w + 1200w^-1 + 180
A' = 60 - 1200/w²
0= 60 - 1200/w²
-60=- 1200/w²
-60=- 1200/w²
-60w²=-1200
w²=20
w=10root2 w = 2root5

 

[TheRandomMultiplyer]

A = (w + 20)(60/w + 6)
A = 60w + 1200/w + 180
A = 60w + 1200w^-1 + 180
A' = 60 - 1200/w²
0= 60 - 1200/w²
-60=- 1200/w²
-60=- 1200/w²
-60w²=-1200
w²=20
w=10root2

I think I got it thanks skeeter

I believe in the second line, there is an error, the 60w should be 6w [for 6*w from both brackets in the previous line]

 

[TheRandomMultiplyer]

I believe in the second line, there is an error, the 60w should be 6w [for 6*w from both brackets in the previous line]

A = (w + 20)(60/w + 6)
A = (60 + 6w + 1200/w +120)
A = (6w + 1200/w + 180)
A = (6w + 1200w^-1 + 180)
A' = 6 + (-1)(1200w^-2)
0 = 6 - 1200w^-2
0 = 6 - 1200/w^2
1200/w^2 = 6
6w^2 = 1200
w^2 = 1200/6 = 200
w = √(200)
w = 10√(2)