The management of the UNICO department store has decided to enclose a 900 ft 2 a

Question

The management of the UNICO department store has decided to enclose a 900 ft2 area outside the building for displaying potted plants and flowers. One side will be formed by the external wall of the store, two sides will be constructed of pine boards, and the fourth side will be made of galvanized steel fencing. If the pine board fencing costs $5/running foot and the steel fencing costs $3/running foot, determine the dimensions of the enclosure that can be erected at minimum cost. (Round your answers to one decimal place.)

Explanation / Answer

Let one side of the fencing be x ft and the other side be y ft

So, x*y = 900; ==> y = 900/x
The three sides fencing can be either (2 sides with x and one side with y) or (2 sides with y and one side with x)

This implies, the fencing length of 3 sides is either (2x + y) or (x + 2y); so there could be four types cost modelling taking into account of the variation in cost of different fencing:

So let us work on each model:

i) (x, y) = ($3, $5) & P = 2x + y

==> C = 6x + 5y; substituting for y from (2), C = 6x + (5*900)/x
Differentiating C' = 6 - (5*900)/x^2;
equating this to zero and solving, x = 27.386 ft (nearly)
Again differentiating, C'' = (10*900)/x^3, which is > 0 for x = 27.386;
Hence cost is least.

Roughly taking x = 27.4 ft, the cost in this case will be nearly $328.63.

ii) (x, y) = ($5, $3) & P = 2x + y

==> C = 10x + 3y = 10x + 2700/x;
==> C' = 10 - 2700/x^2; setting to zero and solving, x = 16.432 ft
Here also C'' at x = 16.432 will be > 0, hence it is least

Roughly taking x = 16.432 ft, the cost in this case also will be nearly $328.63.

iii) (x, y) = ($3, $5) & P = x + 2y
==> C = 3x + 10y = 3x + 9000/x;
==> C' = 3 - 9000/x^2; setting to zero and solving, x = 54.77 ft
Here also C'' at x = 54.77 will be > 0, hence it is least
Proceeding in similar lines as above, x = 54.77 ft and
cost in this case also is nearly $328.64.

iv) (x, y) = ($5, $3) & P = x + 2y
==> C = 5x + 6y = 5x + 5400/x;
==> C' = 5 - 5400/x^2; setting to zero and solving, x = 32.683 ft
Here also C'' at x = 32.683 will be > 0, hence it is least
Proceeding in similar lines as above, x = 32.683 ft and cost
in this case also is nearly $329.54.

As such in any choice, the cost of fencing remain same; so according to the availability of materials and design any one of the four design can be selected.

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