Definitions
Define a planar graph.
A graph with no edges cross each other.
Which variable is plotted on the x axis?
Explanatory variable.
What is the perpetuity formula?
A = M/i
Decide whether this pattern is arithmetic or geometric:
4, 5.58, 7.16, 8.74, 10.32
Bonus [100] if you give the common ratio/difference.
Arithmetic
common difference = 1.58
In a critical path network, what information is placed along the edges?
Activity name/letter
Activity duration
Arrows
What is a confounding variable?
A variable is another variable that has a similar effect on the response variable.
Identify the value of A and B in this least squares regression line:
y = 3.56x + 21.4
A = 21.4
B = 3.56
Write this reducing balance loan as a recurrence relation:
A $450 000 loan is taken at 7.2% p.a. compounding monthly with monthly payments of $850.
A(n+1) = 1.006*An - 850, A0 = 450000
$25 for every ticket bought in the booking.
Show this as a mathematical model.
Determine the cost of a 10 tickets
t(n) = 5 + 25n
let n = 10
t(10) = 5 + 25(10)
t(10) = 255
Using diagram, identify a:
Eulerian Trail [100]
Semi-Hamiltonian Path [100]
Eulerian Trail = A>B>D>E>F>D>C>A
Semi-Hamiltonian = B>A>C>D>F>E
Explain compounding periods.
Compounding periods are the number of times in a year that compound interest is earnt. For example, if compounding monthly, there are 12 compounding periods.
Determine the least squares regression line for the following data * Must show equation*
Rainfall (cm) 14 12 20 24 29
Number of Plants 45 41 49 56 64
Use the equation to predict number of plants with 16 cm of rainfall
y = 1.2856x + 25.5457
let x = 16
y=1.2856 (16) + 25.5457
y=46.1153
Determine the effective interest rate for an investment if 8.56% p.a. compounding quarterly.
i(effective) = 0.0884
= 8.84%
In a geometric sequence, explain how to find the value of r when:
There is a percentage increase
There is percentage decrease
increase = 1 + % (as decimal)
decrease = 1 - % (as decimal)
What is the local time in New York (UTC-5) if it is 8 am Monday in Auckland (UTC +12)
New York is 17 hours behind Auckland
3pm Sunday in New York.
Define a Semi-Hamiltonian Path.
Start and Stops at different vertex and visit each vertex in the network only once.
Predict the amount of fat (y) in a burger when there is 20 grams of protein (x) using the summary statistics:
r = 0.83 sx=14.0 g (x bar) = 17.2 g
sy=16.4 g (y bar) = 23.5 g
y=0.9723x + 6.7764
let x = 20
y=0.9723(20) + 6.7764
y=26.2224 g fat
How much money must be invested to arrange a perpetuity that pays $5000 every quarter at 4.8% p.a.
A = 5000/(0.048/4)
A = $416666.67
Approx = $416667.00
A state forest is used for logging and in a particular block there are 2000 mature trees. Each year the state allows for a set amount of trees to be logged. After the first 3 years a total of 1440 trees were removed. Determine how many more years the plot can be logged before there are no more trees
year 3 = 1440 trees
Y4 = 1440 - 480 = 960
Y5 = 960 - 480 =480
Y6 = 480 - 480 = 0
Therefore there are another 3 years before all the trees are logged over a 6 year period.
Using the table:
Construct a project network and identify the critical path using forward and backward scanning. Times shown are in minutes
Activity A B C D E F
Duration 5 4 7 2 9 10
Prerequisite - A A A B, C, D E
--B,4->
0/0-A,5->5/5--C,7->12/12-E,9->21/21-F,10->31/31
--D,2->
Critical Path = A, C, E, F
Minimum completion time = 31 minutes
Explain float time and provide a formula.
How much time a non-critical activity can be delayed before it impacts the critical path. It is calculate by
FT = LST-EST-AT
Describe the direction and strength of the relationship.
Using mathematical evidence, justify whether using a linear model is suitable.
Direction = negative
Strength = moderate/strong
The coefficient of determination (r2) is used to explain how much of the data can be represented by the line of best fit. r2 = 0.6561 this means 65.61% of the data is explained by the line of best fit. Although a strong correlation, a linear model may not be as suitable for this data set.
Determine the monthly repayments for a $850 000 home loan that has 3.12% p.a. compounding fortnightly for 30 years.
M= $1678.77
A lab is testing a new strand of antibiotics. The petri dish covered in bacteria has an area of 21 cm^2. After the first hour the area covered in bacteria was observed to be 16.8 cm^2 and after the second hour it was 13.44 cm^2.
The lab will deem the antibiotics effective if it can reduce the area to less than 2 cm^ in 10 hours.
Evaluate whether the antibiotics is effective.
r = 1-0.2 = 0.8
T(0) = 21, T(n+1)= 21 * 0.8^(n-1)
At n=10 hours t(10) = 2.82 cm^2
therefore this is greater than 2 cm^2 and would mean the antibiotics is not effective.
Town A (149°E, 28°S) is due west from Town B (153°E, 28°S).
Town C (153°E, 26°S) is due north of Town B. If you live in Town A and the only way to Town C is via Town B, determine the total distance to the nearest kilometre.
Total = 201.24 + 222.4
Total = 423.64
Total = 424 km