determine the dimensions of the enclosure that can be erected at minimum cost

Calculus
Tutor: None Selected Time limit: 1 Day

Jul 17th, 2015

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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.

Please let me know if you need any clarification. I'm always happy to answer your questions.
Jul 17th, 2015

i dont understand what the answer is

Jul 17th, 2015

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