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our hundred eighty dollars are available to fence in a rectangular garden. The fencing for the north and south sides of the garden costs $5 per foot and the fencing for the east and west sides costs $15 per foot. Find the dimensions of the largest possible garden. (Give exact answers.) ft (north side)

Respuesta :

Answer:

Step-by-step explanation:

Available funds = 180 $

Let the rectangle have length l and width w.

Then fencing cost for north and south = 5(2l) and

other two sides = 15(2w)

Total cost = 10l+30w

This is less than or equal to 180

[tex]10l+30w\leq 180\\l+3w\leq 18[/tex]

To have max area we must have l+3w = 18

or l = 18-3w

Area [tex]= lw = w(18-3w) = 18w-3w^2[/tex]

Use derivative test to find maximum area

A'(w) = [tex]18-6w[/tex]

A"=-6

So maximum when I derivative =0 or w =3

Thus dimensions should be w =3 and l = 18-9 = 9

9x3 in feet would be the dimensions for largest garden.

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