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Let one side of the fencing be x ft and the other side be y ft
2) So, x*y = 912; ==> y = 912/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, $6) & P = 2x + y
==> C = 4x + 6y; substituting for y from (2), C = 4x + (6*912)/x
Differentiating C' = 4 - (6*912)/x^2;
equating this to zero and solving, x = 36.986 ft (nearly)
Again differentiating, C'' = (12*912)/x^3, which is > 0 for x = 36.986;
Hence cost is least.
Roughly taking x = 37 ft, the cost in this case will be nearly $296.
ii) (x, y) = ($6, $2) & P = 2x + y
==> C = 12x + 2y = 12x + 1824/x;
==> C' = 12 - 1824/x^2; setting to zero and solving, x = 12.329 ft
Here also C'' at x = 12.329 will be > 0, hence it is least
Roughly taking x = 12.3 ft, the cost in this case also will be nearly $296.
iii) (x, y) = ($2, $6) & P = x + 2y
Proceeding in similar lines as above, x = 73.97 ft and
cost in this case also is nearly $296.
iv) (x, y) = ($6, $2) & P = x + 2y
Proceeding in similar lines as above, x = 24.66 ft and cost
in this case also is nearly $296.
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 37 ft on two sides with steel fencing and one side of about 24.6 ft with pine board fencing may be better for appearance.
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