The management of the UNICO department store has decided to enclose a 929 ft2 ar

Question

The management of the UNICO department store has decided to enclose a 929 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 $4/running foot and the steel fencing costs $2/running foot, determine the dimensions of the enclosure that can be erected at minimum cost. (Round your answers to one decimal place.)

wood side: ft

steel side: ft

Explanation / Answer

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

2) So, x*y = 929; ==> y = 929/x

3) 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)

4) 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) = ($2, $4) & P = 2x + y

==> C = 4x + 4y; substituting for y from (2), C = 4x + (4*929)/x
Differentiating C' = 4 - (4*929)/x^2;
equating this to zero and solving, x = 30.48 ft (nearly)
Again differentiating, C'' = (8*929)/x^3, which is > 0 for x = 30.48;
Hence cost is least.

Roughly taking x = 30 ft, the cost in this case will be nearly $243.87

ii) (x, y) = ($4, $2) & P = 2x + y

==> C = 8x + 2y = 8x + 1858/x;
==> C' = 8 - 1858/x^2; setting to zero and solving, x = 15.24 ft
Here also C'' at x = 15.24will be > 0, hence it is least

Roughly taking x = 15.24 ft, the cost in this case also will be nearly $ 243.84

iii) (x, y) = ($2, $4) & P = x + 2y

Proceeding in similar lines as above, x = 60.96 ft and
cost in this case also is nearly $243.84

iv) (x, y) = ($4, $2) & P = x + 2y

Proceeding in similar lines as above, x = 30.48 ft and cost
in this case also is nearly $243.87

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.

However I feel selecting 30 ft on two sides with steel fencing and one side of about 30 ft with pine board fencing may be better for appearance.

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