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Question

This is an optimization problem. I need help as soon as possible

A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy-duty fencing selling $3 a foot, while the other two sides will use fencing selling for $2 a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced at a cost of $6000.

Answer 1

Ther perimeter will be 2x+2y, which will cost 2*2x+3*2y=6000;
the area will be x*y. Now solve for y in terms of x in the top equation:y=(6000-4x)/6; next substitute this in for the latter equation: Area=xy to get--A=x(6000-4x)/6. Multiply this out to get: A=-(4/6)x^2+1000x. To maximize this you need to complete the square and find the vertex.

Answer 2

Let's say that its dimensions would be x by y, where x would refer to the measure of the sides which require heavy-duty fencing and y would be the measure of the sides which require the other type. Thus, using the given budget, we have and equation relating the costs of fencing

6000 = 2(3x) + 2(2y)
6000 = 6x + 4y

or that would be

y = 1500 - 3/2 x

Since we need to maximize area, we need to maximize the function

A = xy

Replacing y with the derived equation for y from the cost of fencing, we have

A = x (1500 - 3/2 x)
A = 1500x - 3/2 x^2

Deriving and equating to zero [to maximize], we have

0 = 1500 - 3x
x = 500

Thus, y should be

y = 1500 - 3/2 (500)
y = 750

The largest area that could be fenced would then be 500 ft. by 750 ft.

Hope that helps.

Answer 3

Let H represent the length of a heavy-duty side ($3/ft) and L the length of one of the other sides ($2/ft). Then you're solving
max HL
subject to 2(3H) + 2(2L) = 6000

Solve for L and plug the constraint (L=1500-3H/2) into the objective function (HL) to get
max 1500H - 1.5H^2

Take the derivative and set it equal to zero to get
1500-3H=0
which produces H=500, L=750.

Answer 4

Max: A=xy
Const.:
6000= 2(3X)+2(2y)
6000= 6x+4y
divide all by 4,
1500= 1.5x +y
y= 1500-1.5x
substitute this into the area function;
A(x)= (1500-1.5X)*X
= 1500X- 1.5X^2
A'(x)= 1500- 3x..

Answer 5

i donot know