Test2 Key
1. (18 points)
A company manufactures three types of toys A, B, and C. Each requires rubber, plastic, and aluminum as listed below
ToyRubberPlasticAluminum
x1:A224
x2:B122
x3:C124
The company has available 600 units of rubber, 800 units of plastic, and 1400 units of aluminum. The company makes a profit of $4, $3, and $2 on toys A, B, and C, respectively. Assuming all toys manufactured can be sold, determine a production order so that profit is maximum.
a). Set up objective function P and the constraints.
2x1+2x2+2x3 600
2x1+2x2+2x3 800
4x1+2x2+4x3 1400
x1 0, x2 0, x3 0
P= 4x1+3x2+2x3
b). Set up initial simplex tableau, circle the pivot element,
P x1 x2 x3 s1 s2 s3 RHS
0 2 1 1 1 0 0 600
0 2 2 2 0 1 0 800
0 4 2 4 0 0 1 1400
1 -4 -3 -2 0 0 0 0
c) Write all the row operations required to perform one pivot operation.
R1=.5r1
R2=r2-2R1
R3=r3-4R1
R4=r4+4R1
2.(15 points)
Interpret the following tableaus (maximum problems). If it is final stage, state the solution (including values for objective function, basic and non-basic variables). If it requires further pivoting, circle the pivot element. If there is no solution, state the reason why.
P x1 x2 x3 s1 s2 RHS P x1 x2 x3 s1 s2 RHS
I.
Final, x1=4, s1=15, x2=x3=s2=0, P=38 Need to pivot, pivot elemen =1 (row2,
column3
P x1 x2 x3 s1 s2 RHS
III.
No solution, pivot can't be 0 or negative
3. (15 points) Consider the following linear programming problem:
Set up the initial simplex tableau for that problem, circle the pivot element.
(Notice that this is a minimum problem with mixed constraints)
Maximize Z= -C = -2x1-2x2
Z + 2x1+ 2x2=0
Change constraints to:
-2x1 - 2x2 -8
x1-x2 2
-x1+x2 -2
P x1 x2 s1 s2 s3 RHS
0 -2 -2 1 0 0 -8
0 1 -1 0 1 0 2
0 -1 1 0 0 1 -2
1 2 2 0 0 0 0
4.( 10 points)
Jack needs to obtain a $2000 loan that he plans to repay in 2 years.
Bank A offers him 5.4% simple interest loan,
bank B offers him 5.1% discounted loan.
a) Which of the two options is better for Jack?
Answer by computing and comparing the interest Jack will have to pay on each
of the two loans.
A: I=Prt=2000(.054)2=216
B: P=A(1-rt), A=P/(1-rt), A=2000/(1-.051*2)=2227.17
I=A-P=227.17
A is better, less interest
b) After thinking a little Jack decided to borrow $2,000 for only 10 months. He received a simple interest loan from bank C. He paid total of $83 interest on his loan. What was the annual interest rate?
I=Prt
83=200(r)*(10/12)
r=.0498=4.98%
5. (10 points)
Determine the interest earned in one year on an investment of $12,000 at 7.5%
compounded
a). Monthly.
A=12000(1+.075/12)12=12931.59
I=A-P=931.59
b). Daily.
A=12000(1+.075/365)365=12934.51
I=A-P=934.51
c). Briefly explain why one is a little higher than the other.
Daily compounding brings more interest, because interest is computed and added
to the principal more often.
6. (12 points)
Barbara invested $2500 in the bank account that pays 7.4% annual
interest compounded weekly.
a. What is the effective interest rate on that account?
r*=(1+.074/52)52 -1=.0768=7.68%
b. How many years will it take for her investment to double its initial value?
2P=5000
5000=2500(1+.074/52)52t
divide by 2500
2=(1+.074/52)52t
take ln of both sides
ln(2)=52tln(1+.074/52)
t=9.37 years
7. (10 points)
Doug and Pat contribute $1000 per quarter of a year to an IRA paying 9%
annual interest compounded quarterly .
a). What is their total contribution to the IRA after 10 years?
1000*10*4=$40000
b). How much money will they accumulate on their IRA after 10 years?
A=1000 *[ (1+.09/4)40-1]/(.09/4) (annuity formula)
A= $63786.18
8. (10 points)
Warren and Sarah decide to purchase a $200,000 house. They put down
$40,000 and amortize the balance at 8%annual interest compounded monthly
for 30 years.
a). What is their monthly payment?
160000(1+.08/12)360=Pymt[ (1+.08/12)360-1]/(.08/12)
Pymt=$1174.02
b). How much total interest will they pay on the house?
(1174.02*360) - 160000=2622648.39