A fence is to be built to enclose a rectangular area of 230 square feet. The fence along three sides is to be made of material that costs 5 dollars per foot, and the material for the fourth side costs 15 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.

My dimensions that I got is sqrt(215.625)*230/sqrt(215.625) but it's wrong.

cost = 5(2b+L) + 15 L = 10 b + 20 L

bL = 230 so b = 230/L

cost = 2300/L + 20 L

d cost/dL = -2300/L^2 + 20 L

= 0 for min
20 L = 2300/L^2
L ^3 = 115
L = 4.86
b = 47.3
check my arithmetic !!!

cost = 5(2b+L) + 15 L = 10 b + 20 L

bL = 230 so b = 230/L

cost = 2300/L + 20 L

d cost/dL = -2300/L^2 + 20 *******

= 0 for min
20 = 2300/L^2
L ^2 = 115
L = 10.72
b = 21.5
b = 47.3

cost=5*(2w+L)+15L

Area= Lw or w= ARea/L=230/L

cost= 10(230/L)+5L+15L

dcost/dL= -2300/L^2+20=0
L=sqrt 115 and w= 230/sqrt115

To solve this problem, let's consider the dimensions of the rectangular area.

Let's assume the length of the rectangular area is L and the width is W.

The area of a rectangle can be represented as the product of its length and width, so we have:

Area = Length * Width = L * W

According to the problem, the fence is to be built along three sides, which means three sides will have the same cost per foot. The fourth side, however, will have a different cost per foot.

The cost of the fence on the three sides is $5 per foot and the cost of the fence on the fourth side is $15 per foot.

The total cost to build the fence can then be calculated as follows:

Total Cost = Cost of three sides + Cost of the fourth side

Total Cost = [2*(Length + Width)*5] + [Length*15]

Let's simplify this expression:
Total Cost = [10*(Length + Width)] + [15*Length]
Total Cost = 10*Length + 10*Width + 15*Length

Expressing the total cost in terms of a single variable, let's assume Length = x and Width = y.

Total Cost = 10x + 10y + 15x
Total Cost = 25x + 10y

We know that the area of the rectangular area is 230 square feet, so:
Area = Length * Width = x * y = 230

Now we can express y in terms of x:
y = 230 / x

Substituting this value of y in the Total Cost equation, we have:
Total Cost = 25x + 10*(230 / x)

To find the dimensions of the enclosure that is most economical to construct, we need to minimize the Total Cost.

To minimize the Total Cost, we can take the derivative of the Total Cost equation with respect to x and set it equal to zero:

d(Total Cost) / dx = 25 - (10 * 230) / (x^2) = 0

Simplifying this equation, we have:
25x^2 - 2300 = 0

Solving for x, we find:
x^2 = 2300 / 25
x^2 = 92
x = √92
x ≈ 9.59

Now we can find the value of y using the area equation:
y = 230 / x
y = 230 / 9.59
y ≈ 23.96

Therefore, the dimensions of the enclosure that is most economical to construct are approximately:
Length (L) ≈ 9.59 feet
Width (W) ≈ 23.96 feet