Question

Suppose I want to build a garden abutting my house, using the house as the west side of the garden. The south and east sides of the garden will face the street, so there I will use a decorative wall costing $30 per foot. The north side of the garden will use a tasteful fence, costing $20 per foot. If I want a 540 square foot garden, what is the minimum it will cost me?

Answer 1

Minimum cost is 1800 $

Explanation:

Let l be the length of west side and w be the length of north side.

East side = West side = l

North side = South side = w

Area = lw = 540 ft²

We have

`lw=540\\\\w=(540)/(l)`

Cost for fencing

C = 30 w + 30 l + 20 w = 30 l + 50 w

`C=30l+50* (540)/(l)\\\\C=30l+(27000)/(l)`

At minimum cost derivative is zero,

That is

`dC=30dl-(27000)/(l^2)dl\\\\30-(27000)/(l^2)=0\\\\30=(27000)/(l^2)\\\\l^2=900\\\\l=30ft`

We have

lw = 540

30 x w =540

w = 18 ft

Cost = 30 l + 50 w =30 x 30 + 50 x 18 = 1800 $

Minimum cost is 1800 $