Calculus Optimization Question?

Question

A landscape artist is creating a rectangular rose garden to be located in a local park. The garden is to have an area of 60m^2 and be surrounded by a lawn. The surrounding lawn is to be 10m wide on the north and south sides and 3m wide on the east and west sides of the garden. Find the dimensions of the rose garden if the total area of teh garden and lawn together i sto be a minimum.

Answer 1

The area of the rose garden can be represented as such

Ar = l * w = 60

From this we can solve for either l or w to use in the equation for the entire garden.

Ag = [ l + 2(10) ] * [ w + 2(3) ]

2(10) comes from the two additional lengths of 10 meters each.
2(3) comes from the two additional widths of 3 meters each.

It doesn't matter which you solve for in the first equation. I'll solve for l.

l = 60/w

Now we subsitute l with 60/w in the second equation.

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

Ag = 60 + 360/w + 20w + 120

dAg/dw = 20 - 360/w^2

Set it equal to zero.

20 - 360/w^2 = 0

20 = 360/w^2

20w^2 = 360

w^2 = 18

w = 3√2
l = 60/w = 60/3√2 = 10√2

Now add the 6 meters and 20 meters respectively to come up with the dimensions of the entire garden.

w = 3√2 + 6
l = 10√2 + 20

Answer 2

Let east-west length of the garden be x m and north-south length be y m. Then xy = 60. The total E-W length is x + 6 and the total N-S length is y + 10 and we want to minimise
A = (x+6)(y+10) = xy + 10x + 6y + 60
= 60 + 10x + 6y + 60
= 10x + 6(60/x) + 120
dA/dx = 10 + 360(-1)/x^2
= 0 when 360/x^2 = 10, so x^2 = 36, i.e. x = 6 (-6 is not meaningful).
d2A/dx2 = 10 - 360(-2)/x^3
This is positive for x = 6, so this is a minimum.
Hence the garden should be 6m E-W and 10m N-S.

Answer 3

Let x = length of rose garden.
Then x+20 = length with lawn
Width of rose garden =A/length = 60/x
Then 60/x +6 = width with lawn
Total area with lawn =
At = (x+20)(60/x +6)
=60 +6x +1200/x +120 = 6x + 1200/x +180
dAt/dx = 6 -1200/x^2 Set dAt/dx to 0 to find minimum
1200 = 6x^2
x^2 = 200
x = 10sqrt(2) meters = length of rose garden
60/x = 60/10*sqrt(2) = 3sqrt(2) meters = width of rose garden