Question

A gardener wishes to design a rectangular rose garden with one side of the garden along a main road. The fencing for the 3 non-road sides is $6 a foot and the fencing for the 1 road side is $8 a foot. A further condition is that the total cost for the fencing must come to $2 for every square foot of the area of the rose garden. What is the minimum cost for the fencing that can satisfy these conditions?

ANY HELP WOULD BE GREATLY APPRECIATED

Answer 1

This problem does not have solution.

When you do the algebra you find that the statementes lead to the equation of a hyperbola which does not have a minimum. And so there is not a minimum cost.

This is how you may get to that conclusion using math:

1) variables:

x: length of the side of the fence parallel to the road
y: length of side of the fence perpendicular to the road

2) area of the garden enclosed by the fence: xy

3) cost of the fence: multiply each length times its unit cost per foot

cost = 6x + 8x + 6y + 6y

cost = 14x + 12y

4) cost is also equal to $ dolars times the area = 2xy

So, 2xy = 14x + 12y

=> 2xy = 14x + 12y

Also, do not forget that x and y has to satisfy x>0 and y>0


You can solve for y (or x if you prefer)

2xy-12y=14x
xy - 6y = 7x
y(x-6)=7x
y = 7x / (x -6)

You can verify that as you increase x (starting any x > 6 to make y positive) to make y minimal, y will decrease but the produc xy will increase.

for example do x = 100, you get xy ≈ 744, x = 1000 xy≈7042, and that trend never ends.

If you know about limits you can show that.

At the end, there is not a minimum cost

Answer 2

There is no minimum cost for the fencing that can satisfy these conditions.

as the equation obtained for this problem is y =
`(7x)/(x-6)`.

Explanation:

let length of the rectangular rose garden be x

and breath of the rectangular rose garden be y

then the area of the garden = xy

cost of fencing a foot for the 3 non road sides = $6

and cost of fencing a foot for a road side = $8

total cost of fencing = 6x + 8x + 6y + 6y

so, total cost = 14x + 12y

cost of fencing for every square foot of the area =$2

So, 2xy = 14x + 12y

=> 2xy = 14x + 12y

Also, x and y has to satisfy x>0 and y>0

You can solve for y (or x if you prefer)

2xy-12y=14x

xy - 6y = 7x

y(x-6)=7x

y =
`(7x)/(x-6)`

we can check that this function has no minimum value.

as we increase x (starting any x > 6 to make y positive) to make y minimal, y will decrease but the product xy will increase.

So, there is not a minimum cost for fencing that can satisfy these conditions.