A rectangular building is supposed to be 2,000 cubic ft in volume. it is designe

Question

A rectangular building is supposed to be 2,000 cubic ft in volume. it is designed so that the front is all glass. The other 3 outside walls are block. The floor is concrete and the roof is flat.

The cost of the "Glass wall" is 24$ per square foot.

The cost of the block walls is 8$ per square foot

The cost of the roof is 10$ per square foot

The cost of the floor is 6$ per square foot.

Using Lagrange multipliers or 2nd order partial derivitive test, give the dimensions that will MINIMIZE the cost.

Using either Lagrange Multipliers or 2nd order partial test, justify the value you found is infact a minimum.

Show all dimensions to the nearest foot.

I have included a picture of what the bulding looks like from the book.

Explanation / Answer

Let x,y,z be as in the figure.

Constraint is xyz=2000........................(1)

Cost function

                     C(x,y,z) = 24xz+24yz+16xy..............(2)

From (1) and (2) , we need to minimize the function F(x,y) of two variables

                    F(x,y) = 24x2000/y+24x2000/x+16xy

                               =8[6000/y+6000/x+2xy]

To find the minimum value , set

Fy = 8[-6000/y2 +2x] =0

Fy = 8[-6000/y2 +2x] =0

In view of the symmetry, x =y and so x3 =3000, giving x =y=14.42 and z =9.615 (all in ft)

To confirm it is actually a (the) minimum , consider the second partial derivatives (values at the critical point)

Fxx =Fyy = 8x12000/x3 = 24>0

and FxxFyy -Fxy2 = (24)2- (16)2 >0.

These coniditions ensure that the value is minimum

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.