MATH 1100: QUANTITATIVE ANALYSIS
TEST #1 (VERSION A)
1. By calculating the reduced form, find all solutions to
x + 2y − z = 3
3x + y = 4
2x − y + z = 2.
Solution: There are no solutions.
2. Using matrix inverses, solve:
4x − y = 1
x + 2y = 2
Solution:
4 −1
1 2
−1
=
2
9
− 19
1
9
4
9
and so
x=
4
7
,y = .
9
9
3. Suppose that $1000 is invested at 10% nominal interest rate, compounded
continuously.
(a) Write an equation describing the value of the investment after t years
since the principal was invested.
(b) Approximate the value of the investment after 30 years, using the
approximation e ∼ 3.
(c) How many years does it take for the investment to double in value?
Hint: use the approximation ln(2) ∼ 0.7.
Solution:
(a)
S = 1000e0.10t
(b)
S = 1000e0.10×30
= 1000e3
∼ 1000 × 33
= 1000 × 27
= 27000.
Date: September 3, 2001.
1
2
MATH 1100: QUANTITATIVE ANALYSIS
TEST #1 (VERSION A)
300
200
100
0
100
200
300
400
A
–100
Figure 1. Beside the two axes, there are two other constraints
(c)
0.7
ln(2)
=
= 7 years.
r
0.1
4. Your company makes two products, A’s and B’s. The A’s require 2 hours
of work each to make, while the B’s require 4. You have 800 work hours
available each day to make them. But you can only transport a total of
300 products (A’s and B’s together) in a day. A’s sell for $30, B’s for $40.
Suppose you will always sell as many as you make. How many of each
should you make to maximize sales revenue?
Solution: See the picture in figure 1. The constraints are
t=
2A + 4B ≤ 800
A + B ≤ 300
A≥0
B≥0
This gives the part of the upper right quadrant which is beneath both of
the lines 2A + 4B = 800 and A + B = 300. The revenue is R = 30A + 40B.
The maximum is at 200 A’s, 100 B’s.