a woman wants to construct a box whose base length is twice the base width. The material to build the top and bottom is $ 9/m ( squared) and the material to build the sides is $6m/ ( squared). If the woman wants the box to have a volume of 70 m cubed, determine the dimensions of the box ( in meters) that will maximize the cost of production. What is the minimum cost?
To determine the dimensions of the box that will maximize the cost of production, we need to express the cost function in terms of the base width (w) and then find its maximum value.
Let's assume the base width of the box is w. According to the problem, the base length is twice the base width, so the base length (l) would be 2w.
The height of the box (h) can be found by dividing the volume of the box by the base area. Since the volume is given as 70 m^3, and the base area is length times width (l * w), we have: h = V / (l * w) = 70 / (2w * w) = 35 / w^2.
Now, let's find the expressions for the surface areas of the box:
The top and bottom surfaces have the same dimensions as the base, so their areas would be l * w = (2w) * w = 2w^2.
The three sides of the box can be divided into two pairs of rectangular faces. Each pair would have dimensions l * h and w * h. So the total area of the sides would be: 2lw * h + 2wh * h = 2lw * h + 2wh * h = 2lwh + 2wh^2 = 2w(2w)(35 / w^2) + 2w(35 / w^2)^2 = (140 / w) + (2450 / w^2).
Now, let's express the cost function. Since the top and bottom materials cost $9/m^2, and the side materials cost $6/m^2, the cost (C) of production can be expressed as:
C = (2 top/bottom areas * 9) + (side area * 6)
C = 2(2w^2)(9) + ((140 / w) + (2450 / w^2))(6)
C = 36w^2 + 840/w + 14700/w^2.
Now, we need to find the maximum value of the cost function by taking the derivative of C with respect to w and setting it equal to 0:
dC/dw = 72w - 840/w^2 - 29400 / w^3 = 0.
To solve this equation, we can multiply through by w^3 to get rid of the fraction:
72w^4 - 840 - 29400 = 0.
Simplifying further:
72w^4 = 30240,
w^4 = 420.
Taking the fourth root of both sides:
w = ∛(420) ≈ 7.07.
Since the box dimensions cannot be negative, we discard the negative root.
So, the base width of the box is approximately 7.07 meters, and the base length would be twice the width, so approximately 14.14 meters.
To find the height, we can substitute the value of w into h = 35 / w^2:
h = 35 / (7.07)^2 ≈ 0.697 meters.
Therefore, the dimensions of the box that will maximize the cost of production are approximately: width = 7.07 meters, length = 14.14 meters, and height = 0.697 meters.
To find the minimum cost, substitute these values into the cost function:
C = 36(7.07^2) + 840/7.07 + 14700/(7.07)^2
C ≈ $1085.13.
So, the minimum cost of production for the box is approximately $1085.13.