School of Business: BBA120 Business Mathematics Assignment 2019

This assignment solution addresses several key concepts in business mathematics. Question one analyzes student preferences for TV shows (Game of Thrones, Prison Break, and 24) using set theory to determine the number of students who like at least one show, at most one show, and specific combinations of shows. Question two involves population growth calculations based on a given growth rate and initial population, and the determination of the year in which the population reached a specific value. Question three explores investment scenarios, comparing returns from Barclays and Standard Chartered over time, including calculations for compounded interest and the use of matrices. Question four deals with demand and supply analysis, solving for equilibrium price and quantity, calculating breakeven points, and constructing profit functions. Finally, question five focuses on revenue optimization, determining the optimal production level to maximize revenue, and analyzing cost, demand, profit, and marginal profit functions using quadratic equations and calculus principles.

Question one
A. TV shows
I. Like at least one
This mean the number of students who like one TV shows and above
The solution is 43+ 44+24 ¿=111
II. At most one
This means the number which like a total of one show at most
Game of thrones only 26
Prison Break only 24
24 only 9
No TV show liked 39
Hence at most one (one or zero shows) gives 98 students
III. Game of thrones or 24 only
Game of thrones only 26
24 only 9
Total 35
IV. Prison break and 24
8 students
V. Game of thrones only 26 students.
B. Taking apple juice only 25
Taking orange juice only 75
Taking both apple and orange 75
Total 175
Taking neither of the juices 225 students
Question two
A. Population follows; p ( t ) =20 ert
P is the population in thousands, t the number of years since 1980 and r the growth rate
which is 10%.
I. As at 1999
1999−1980=19

p ( 19 )=20 e(0.1∗19)=133.7179
Hence P is 133,718
II. In 2020
t=2020−1980=40
p ( 40 ) =20 e(0.1∗40)=1091.9630
P is 1,091,963
III. The population was 105000 in which year
p ( t ) =20 ert =105000
20 ert= 105000
1000 =105
20 ert=105
ert=5.25
e0.1 t =5.25
0.1 t=ln 5.25
t=16.58
The year is ( 1980+16.58 ) =1996.58
Which is 1996
IV. p ( 35 )=20 e35 r=182000
20 e35r =182
e35 r=9.1
35 r=ln ( 9.1)
r = ln ( 9.1 )
35 =0.06309=6.31 %
B. Solve for x
log5 ( x−10 ) −log5 3=2
That means
log5 ( x−10
3 )=2
52= ( x−10
3 )

25=( x−10
3 )
75=x−10
x=85
C. 7 e3 x−5 +7.9=47
7 e3 x−5=39.1
e3 x−5= 39.1
7
3 x−5=ln ( 39.1
7 )
x=2.2401
Question three
A. Investment
δ=ln (1+i)
Barclays
δ=ln ( 1.132 ) =0.12399
Standard Chartered
δ=ln ( 1. 0 3 5 )=0.0344
I. At time t
Amount in Barclays is
A=7500 e0.12399 t
Amount∈ Standard Chartered is
A=11000 e0.0344 t
Equating the same
7500 e0.12399t =11000 e0.0344 5 t
e0.12399t = 11000e0.0344 t
7500
e0.12399t
e0.0344t = 11000
7500

e0.0 8959t = 22
15
t=4.274944215 years
II. Quadruple gives (7500∗4 ¿=30000
Number of quarters 38
30000=7500 ( 1+ r
4 )
38
1.0372=1+ r
4
r =14.88 %compounded quarterly
III. After 62.4 months
Barclays
A=7500∗e0.12399∗5.20=K 14,291.30
Standard Chartered
A=11000 e0.0344∗5.2
A=k 13,154.65
B. 2 x+ y =7
x−9 y =13
Matrix
[ 2 1
1 −9 ]∗[ x
y ]= [ 7
13 ]
The inverse of [2 1
1 −9 ]=
[ 9
19
1
19
1
19
−2
19 ]Now
[ x
y ]=
[ 9
19
1
19
1
19
−2
19 ]∗
[ 7
13 ]=[ 4
−1]
x=4 ,∧ y=−1
Question four

A.
I. Demand
Price 1
Quantity 90
Price 2
Quantity 30
Supply
Price 1
Quantity 20
Price 2
Quantity 100
Let demand be x and supply y
Then
90 x +20 y=0
30 x+ 100 y =0
II. Solving the simultaneous equation gives
Price K4.2
III. From the calculation above
Breakeven quantity is 50
B.
I. Selling price K60
Fixed cist K18000
Variable cost ¿ 0.25∗60 x assuming the firm produces x units
This gives 15 x
Profit function
TR−TC
TR=60 x
TC=15 x+18000
p ( x )=60 x−15 x+18000
p ( x )=4 5 x +18000
II. Average profit function

p ( x )
x = 45 x+ 18000
x =45+ 18000
x
Question five
A. Demand p=6+1
2 q
Revenue; (6+ 1
2 q )q= 1
2 q2 +6 q
At optimum point
q+ 6=0
Hence at optimum revenue production need to be 6 units
B. Demand function; p=100−0.01 q
Cost function; c ( q ) =50 q+100
I. Marginal profit function
TR=q(100−0.01 q)
¿ 100 q−0.01 q2
TC=50 q+100
Profit function; TR−TC
100 q−0.01 q2−(50 q+100)
50 q−0.01 q2−100
Marginal profit; 50−0.02 q
II. Marginal cost
TC =50 q+100
Marginal cost; ¿ 50
III. 50 q−0.01 q2−100=700
−0.01 q2 +50 q−8 00=0
Using the quadratic formula
x=−b ± √ b2−4 ac
2 a
q=−50 ± 49.68
−0.02

at q=16units
C. MR=40−10 q2
I. TR=40 q−10 q3
3 + c
Where c is the fixed cost
II. Demand function
demand fuction=TR/q
40 q− 10 q3
3 + c
q =40− 10
3 q2 + c
q
Where c is the fixed production cost