Math 164: Homework #1, due on Wednesday, April 8
No late homework accepted.
Reading: Chapter 1 (sections 1.1-1.5)
[1] (Fitting a quadratic function to data) The following points in the plane
are assumed to lie on the graph of a quadratic function (t, b(t)). These
points, denoted by (ti, bi), have the coordinates (0,1), (2,7), and (5,46).
Find the quadratic function and plot its graph.
[2] A manufacturer of office furniture is trying to maximize the monthly
revenue of the factory. Various orders have come in that the company could
accept. They include desks, bookshelves, cabinets with doors, and cabinets
with drawers. The table above indicates the quantities of materials and
labor required to assemble the four types of furniture, as well as the rev-
enue earned. Suppose that 6000 units of wood and 2000 units of labor are
available. Formulate the linear programming model that will maximize the
revenue under the given conditions, where xi,i= 1,2,3,4 is the number of
pieces of furniture to be produced for each type (see section 1.3)
Piece Labor Wood Revenue
desk 8 12 200
bookshelf 6 10 100
cabinet with doors 2 25 150
cabinet with drawers 4 20 200
[3] Four buildings are to be connected by electrical wires. The positions of
the buildings are as follows: the first building’s shape is an ellipse with center
(0,0) and horizontal and vertical axes 0.5 and 0.6. The second building is
a disk with center (4,0) and radius 1. The other two buildings are squares
centered at (0,4) and at (4,4), with sides parallel with the axes and of length
2. The electrical wires will be joined at some central point (x0, y0), and will
connect to building iat position (xi, yi).
(a) Plot the positions of the three buildings in the plane.
(b) Formulate the non-linear optimization problem that minimizes the
amount of wire used (see section 1.5)
[4] Assume that mpoints (ti, bi) are given in the plane, i= 1,2, ..., m. For-
mulate a least-squares minimization problem in the unknown x= (x1, x2, x3, x4, x5),
for fitting a curve defined by b(t) = x1+x2ex3t+x4ex5tthrough the points
in an optimal way (see section 1.4)
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