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 $

Step-by-step 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

                   $lw=540\\\\w=\frac{540}{l}$

 Cost for fencing

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

              $C=30l+50\times \frac{540}{l}\\\\C=30l+\frac{27000}{l}$

At minimum cost derivative is zero,

That is

                $dC=30dl-\frac{27000}{l^2}dl\\\\30-\frac{27000}{l^2}=0\\\\30=\frac{27000}{l^2}\\\\l^2=900\\\\l=30ft$

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 $