Here is the problem I am working on:
Romeo is commissioning the construction of a four-inch-tall box made of precious metals to give to his bride, Juliet. The jewelry box will have rectangular sides, a bottom and a top.
The longer sides and the top will be made of gold, at $300/in^2.
The shorter sides will be made of platinum, at $550/in^2.
The bottom will be made of plywood, at $0.02/in^2.
What dimensions will provide Romeo with the lowest cost if he is adamant that the box have a volume of 50 in^3?
I am aware that this involves minimizing the cost, and thus taking the derivative of a cost function for the situation. Can anyone tell me what the cost function for this would be?
Thanks!
Let x=long side, y=short side, z=height
(note: since the cost of material for the short side is more expensive, the minimization will automatically result in y being the shorter side without introducing an additional constraint of x-y>0, but do check when you get your answers)
The cost function would be:
C(x,y,z)
=300xy+2*300xz+2*550yz+0.02xy
=300.02xy+600xz+1100yz
subject to xyz=50.
I get $23,731.91 as the cost.
For your checking purposes, I get 12.64 inches as the sum of x,y and z.
Other Lagrange multiplier examples are available at this site at:
http://www.jiskha.com/display.cgi?id=1289839685
http://www.jiskha.com/display.cgi?id=1291776389
I think MathMate did not catch the fact that the box has to be 4 inches high, thus eliminating the variable z.
Cost = 300(8x+xy + 550(8y) + .02xy
= 2400x + 300xy + 4400y + .02xy
but V = 4xy
4xy = 50
y = 50/(4x)
cost = 2400x + 300x(50/4x) + 4400(50/(4x) + .02x(50/(4x))
= 2400x + 3750 + 55000/x + .25
d(cost)/dx = 2400 - 55000/x^2 = 0 for min of cost
2400 = 55000/x^2
x^2 = 55000/2400 = 22.916666
x = 4.79
y = 50/(4(4.79)) = 2.61
the box should be 4.79 in long, 2.61 in wide, and 4 inches high
Thanks Reiny!
To find the cost function for this problem, we need to consider the cost of each material used.
Let's assume that the dimensions of the box are as follows:
- Length (L) of the longer sides and the top
- Width (W) of the shorter sides
- Height (H) of the box
The volume of the box is given as 50 in^3: L * W * H = 50.
Now let's determine the cost function by calculating the cost of each material.
The cost of the longer sides and the top, made of gold, can be calculated as:
Cost of gold = Area of longer sides and top * Cost per square inch of gold
= (2L * H + L * W) * $300
The cost of the shorter sides, made of platinum, can be calculated as:
Cost of platinum = Area of shorter sides * Cost per square inch of platinum
= 2W * H * $550
The cost of the bottom, made of plywood, can be calculated as:
Cost of plywood = Area of bottom * Cost per square inch of plywood
= L * W * $0.02
Now we can find the total cost function by summing up the costs of all the materials:
Cost = Cost of gold + Cost of platinum + Cost of plywood
= (2L * H + L * W) * $300 + 2W * H * $550 + L * W * $0.02
So, the cost function for this problem is:
Cost(L, W, H) = (2L * H + L * W) * $300 + 2W * H * $550 + L * W * $0.02
To find the dimensions that minimize the cost, we need to differentiate the cost function with respect to each variable (L, W, H), set the derivatives equal to zero, and solve for the variables.