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Mathematics Applied 2012Mathematical Modeling Test

Name:

Criteria to be assessed:

1 Communicate mathematical ideas and information.

4 Use algebraic or graphical linear and non-linear models to solve.

Instructions

Attempt all questions

Working, presentation and explanation of solutions will be taken into account

when marking.

If a question requests algebraic solutions, then they must be shown for full

marks.

TQA Information Sheet provided

CAS calculator provided by student

Time Permitted: 1 hour 30 minutes

Attempt all questions

Maths Applied 3 – Algebraic Modelling Test 2012

Question 1

Jackson sells disposable raincoats at the football. Jackson records the rainfall each week, and the number of raincoats he sells, in order to be to predict how many raincoats he needs to bring to each match.


(a) Prepare a screen plot of the data, and include a labelled sketch below.

(b) Determine the linear relationship between the rainfall, r and the number of raincoats sold, n. Give your numbers to two decimal places.

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t C

t C

Rainfall, (r)mm

No. of raincoats sold (n)

20 9022 12010 508 4525 13027 12021 11012 740 010 70

Question 1 continued.

(c) (i) From the equation, what is the value of the y-intercept? Explain whether this is realistic for this situation.

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(ii) What is the value of the gradient? State the units and explain its meaning in terms of the question....................................................................................................................................................

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(d) Plot a residuals graph on the grid provided below and label 2 points

(e) Use the residuals plot to discuss the validity, or otherwise, of the linear model....................................................................................................................................................

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t C B

t C B

t C B A

t C

Question 2

The Caitlyn Dunn real estate company pays its agents according to the table below:


(a) Complete the grid of pay scales below for sales up to $1,000,000.

Fill in each of the coordinate points and label and scale both axes.

(b) Find the equations for wages earned for sales between

(i) $0 - $200,000 (ii) $800,000 – $1,000,000

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t C B

(200000, )

(400000, )

(0, )

(800000, )

(1000000, )

t C B

Sales (s) Wage (w)$0 - $200,000 $30,000

$200,000 - $400,000 $30,000 + 10% of sales over $200,000

$400,000 - $800,000 $50,000 + 22% of sales over $400,000

$800,000 and above $138,000 + 35% of sales over $800,000

Question 2 continued

The global financial crisis has led the Caitlyn Dunn real estate company to look to reduce costs by implementing a new pay scheme, which will pay agents 20% of their total sales.

(c) Write an equation for an agent’s wage (w) for sales (s) up to $1,000,000.

Represent this on the grid in part (a).

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(d) Algebraically determine the levels of sales where the agents will earn the same wage under both schemes?

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For what levels of sales would the Caitlyn Dunn real estate company be well advised to implement the new scheme to save on costs?

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t C

t C B A

Question 3

The earthquake and tsunami in Japan one year ago caused major problems within the Fukushima nuclear power plant. The reactor overheated and exploded. One of the radioactive elements released was iodine. A nearby spinach crop was dusted with this radioactive substance.

Since the explosion, scientists have been measuring the level of radioactivity in radiation units (r) in the crop each month.

They obtained the scatter-plot and residual plots below.

(a) Explain what type of non-linear equation has been used to model the data and why it was chosen.

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(b) The point (5, 8830) is on the scatter plot. Interpret this point in terms of the variables being investigated.

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(c) The point (5, -530) is on the graph of residuals. Interpret this point in terms of the variables being investigated.

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(5, 8830)

Radiation units (r)

Months since explosion

Months since explosion

Residuals

t C

t C B A

t C B

(5, -530)

Question 4

Abbie is selling hamburgers, made from local beef, at Agfest this year. she sells each burger for $4 each. The ingredients for each burger costs $2. Renting a stand at Agfest costs $100.

(a) Write equations for the revenue (R) and cost (C) functions for 0 - 80 hamburgers....................................................................................................................................................

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Abbie is happy to have sold 80 hamburgers. It is late in the day, so she looks to sell the rest of the hamburgers quickly by reducing the price to $1 each. Abbie sells another 20 hamburgers at this reduced price.

(b) Write the revenue function for 80 – 100 hamburgers....................................................................................................................................................

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(c) Graph the cost and revenue equations for 0 – 100 hamburgers below.


t C

t C B

t C B

Question 4 continued.

(d) Use your graph to estimate the number of hamburgers Abbie must sell to break-even. Show this on the graph....................................................................................................................................................

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(e) (i) Use your graph to determine the number of hamburgers for maximum profit.

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(ii) Determine the maximum profit algebraically....................................................................................................................................................

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t C B

t C B

t C B

Question 5

In 1969 the Hydro Electric Commission dammed the Forth River to create Lake Barrington, in Tasmania’s North-West.The data below shows the height above the original river level as the volume of the lake increases.


t C B A

A scatterplot of the data is shown below:

0 50 100 150 200 250 300 350 4000

20406080

100

120140

160180

Volume, V (Megalitres)

Height above river level, H

(metres)

(a) Explain which is the best relationship to model the data. Write the equation below expressing numbers accurate to 2 decimal places...................................................................................................................................................

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(b) (i) Find the correlation coefficient for the relationship and interpret it in terms of the

data.

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Volume, V (megalitres)

Height above river level, H

(metres)0 0

50 43.20100 80.48150 90.17200 121.12250 120.39300 148.50350 158.27400 165.72

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(ii) Find the coefficient of determination for the relationship and interpret it in terms of the data..............................................................................................................................................

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(c) Using your equation, predict the height above the river-bed when 100 mega-litres of water had entered the lake.

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(d) The engineers estimated that the lake would be full when it reached a depth of 190 metres. Use your equation to predict how much water was needed to fill Lake Barrington.


t C B

t C

t C B

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(e) Comment on the reliability of your answers in (e) and (f)

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Spare graphs and working space

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t C B

t C B

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(200000, )

(400000, )

(0, )

(800000, )

(1000000, )

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