Posted by **tiana** on Sunday, January 2, 2011 at 2:20pm.

i understand optimization but im stuck on this problem.. mrs.day is building a new deck. she has 580 square feet to enclose. if one side is bounded by a wall, find the minimum cost to build the deck if one pair of opposite sides cost $12 per foot and remaining sides cost $19 per foot. what are the demisions of the deck?

- calculus -
**Damon**, Sunday, January 2, 2011 at 2:36pm
Is she building a deck or a railing around the deck.

From the wording I will assume a railing.

The two sides of the railing, length a, cost 12/ft each.

The front of the railing, length b, costs 19.ft

a*b = 580

2*12 a + 19 b = c, the cost

dc/da = 0 for min = 24 da/da +19 db/da

but

b = 580/a

db/da = -580/a^2

so

0 = 24 +19 (-580)/a^2

24 a^2 = 19*590

a^2 = 459

a = 21.4

then b = 27

Now it might be that the front cost 12 and the sides cost 19

in that case

2*19 a + 12 b = c

dc/da = 0 = 38 da/da + 12 db/da

db/da is still -580/a^2

38 = 12(580)/a^2

a^2 = 183

a = 13.5

then b = 42.8

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