Math 3191 Extra Homework #1

1. A manufacturer wishes to melt and combine some or all of six alloys that are available from a distributor to produce 100 pounds of a new alloy that is (by weight) 6% gold, 15% silver, and 30% iron. The six available alloys and their percentage content of gold, silver, and iron are given in the table below.

[Table of percentages]

Prepare an order for the distributor that will allow you to produce the required alloy.

2. Solve the following system of equations.

                    (a + b + c)^2 + 5(a + b +c) - 2 = 0
                                      3a - 2b + 7c  = 0
                                       5a + 4b + 2c = 0
3. Determine if [Matrix 1]can be transformed into[Matrix 2] by elementary row operations.

4. Find a solution of the following system of equations such that x, y, z and w are integers.

                            2x + y - z + w = 5
                           3x - 2y + z - w = 0
                          2x + 2y - 2z + w = 3
5. A grain silo typically has an irregularly shaped hopper at the base. At inventory time, someone stands at the top of the silo and drops a tape down the interior to the surface of the grain. The reading (in feet) is reported to the manager who consults a table (provided by the builder) that gives the number of cubic feet of grain remaining in the silo based on the distance from the surface of the grain to the top of the silo. An actual table for a silo owned by Stillwater Milling Company is given below.

[Table of volumes ]

We want to find a function C(x) that gives the cubic feet of grain in terms of the distance x from the surface of the grain to the top of the silo. The data points 0 through 40 correspond to the regular cylindrical portion of the silo, and thus are described by a linear function f(x) = ax + b for a suitable choice of a and b. The last 10 data points for depths 41 through 50 cannot be accurately approximated by a linear function, so the idea is to find a polynomial function g(x) that does. When we do, we get

[Function Definition ]

(a) Find f(x).

(b) There are a number of ways to find a function g(x) that approximates the last 10 data points. We will simply use a fourth degree polynomial that passes through the data points 41, 43, 45, 47, and 50, and check that the function doesn't miss the remaining points at depths 42, 44, 46, 48, and 49 feet by very much. [More sophisticated methods for doing this approximation will be discussed later in the course.]

If g(x) = ax^4 + bx^3 + cx^2 + dx + e fits the data points for depths 41, 43, 45, 47, and 50 feet, write a system of five equations in five unkowns that must be solved to determine the coefficients a, b, c, d, and e.

(c) Find g(x).

(d) For depths 42, 44, 46, 48, and 49 feet, calculate g(x) and the error.

(e) Find the approximate amount of grain in the silo when the depth is 44.5 feet.