**Submission instructions:**

You will find instructions for completing TMAs in the ‘Assessment’ area of

the MU123 website. Please read these instructions before beginning work on this TMA.

Reviewing your tutor’s comments on your previous TMA will help you as

you work on this one.**MU123 Discovering Mathematics Assignment-The Open University UK**

**Special instructions**

Remember that you need to explain your reasoning and communicate your

ideas clearly, as described in Subsection 5.3 of Unit 1. This includes:

• explaining your mathematics in the context of the question

• the correct use of notation and units

• appropriate rounding.

Your score out of 5 for good mathematical communication (GMC) will be

recorded against Question 8. You do not have to submit any work for

Question 8.

**Question 1**

This question is based on your work on MU123 up to and including Unit 10.

(a) This part of the question concerns the quadratic equation

**Question 2**

This question is based on your work on MU123 up to and including Unit 10.

Aikta is the business owner of a small company which manufactures organic smoothies for local shops and restaurants. Because her product is organic, it has a short life span, so she needs to ensure that she is not over-producing, as waste product has a negative impact on profit margins. She decides to create an economic model using a computer program to predict how much smoothie should be produced in each production run (batch) to maximise profit.

**MU123 Discovering Mathematics Assignment-The Open University UK**

She is aware that if too little smoothie is made, the company will not meet

consumer demand and will miss the opportunity to sell more. If too much

smoothie is made, there will be an excess of unsold smoothie, which will go

bad, while still incurring the production cost to the company.

Aikta’s computer program models the situation as a quadratic equation of

the form y = ax 2 + bx + c,

where y represents the profit (in thousands of pounds) the company is

predicted to make, and x represents the amount of smoothie (in thousands of litres) produced in each batch. The values of a, b and c are determined by

the computer code that Aikta writes, and depend on various economic

measures and results of market research which Aikta inputs manually before each batch of smoothie is produced.

(a) Aikta wants to check that her code is working correctly. She inputs

some test values and runs the code to simulate the model. The quadratic equation produced is

y = 4.862x 2 − 7.211x + 4.836.

By considering the coefficient of x 2, explain why Aikta’s code must contain an error.

(b) Aikta corrects the error, and runs the program again. This time the

quadratic equation produced is

y = −7.291×2 + 5.771x − 0.502.

In this part of the question, you are asked to consider the parabolic

graph modelled by the equation y = −7.291x + 5.771x − 0.502.

(i) Explain mathematically why the y-intercept is −0.502.

(ii) Using the context of the question, give one possible reason why the

y-intercept would be negative.

(iii) Use the quadratic formula to find the x-intercepts. Give your

answers correct to three decimal places.

(iv) According to Aikta’s model, what is the minimum amount of

smoothie the company needs to produce in order to break even

(i.e. make no profit and no loss)? Explain your answer.

(v) The company sets a target of making £600 profit in the next batch.

The company decides to produce 500 litres of smoothie (i.e.

0.5 thousand litres of smoothie). According to Aikta’s model, will

the company exceed or fall short of its target? Explain your answer.

**MU123 Discovering Mathematics Assignment-The Open University UK**

**Question 3**

This question is based on your work on MU123 up to and including Unit 11.

Joe works for a small company which employs 20 people. He is interested in

investigating the effect of the COVID-19 pandemic on the number of hours

worked by the company’s employees. To get an initial sense of what the

effect might be, he picks the first full working week in June 2019 and

compares it with the equivalent week in June 2020. For each of the two

chosen weeks, he downloads the number of working hours logged by each

employee. The raw data are shown in Table 1.

Table 1 Number of hours worked in the first full working week of June

(a) Joe considers displaying the data as a pair of dotplots. Do you think

this is a good idea for these particular datasets? Give a brief reason for

your answer.

(b) Joe decides to create a box plot for each year as displayed in Figure 2.

With reference to Subsection 1.2 of Unit 11, state three ways in which

Joe could improve the clarity of his presentation of the boxplots.

(c) Joe can’t remember which boxplot belongs to which year. State one way

in which he could check by looking at the raw data for each year.

(d) Joe realises that the top boxplot represents the data for 2020 and the

bottom boxplot represents the data for 2019. He tries to interpret what

the data is telling him by making the statements below. For each one,

state whether you agree or disagree with Joe’s interpretation, and

explain why you think he is correct or incorrect.

(i) ‘On average, employees worked more hours in 2020 than in 2019.’

(ii) ‘There was greater variability in the number of hours worked in 2020 compared with 2019.’

(iii) ‘More than half of employees in 2020 worked more than the median

number of hours.’

(e) Use the boxplot for 2020 to say whether the data are symmetrical or

skewed. If the data are skewed, then state whether they are skewed to

the left or skewed to the right, explaining your reasoning briefly.

(f) Joe creates (poorly presented) histograms for each of the two datasets,

shown in Figure 3 and Figure 4 below.

(i) Which histogram belongs to the data for 2020 – Figure 3 or

Figure 4? Give a reason for your answer.

(ii) Comment on one aspect of the comparison of data from the two

years that can be seen more easily on the histograms than on the

box plots.

(iii) Comment on one aspect of the comparison of data from the two

years that can be seen more easily on the boxplots than on the

histograms.

**MU123 Discovering Mathematics Assignment-The Open University UK**

**Question 4**

This question is based on your work on MU123 up to and including Unit 12.

(a) Figure 5 shows a sketch of a triangle, where all side lengths are

measured in centimetres. Find the length of the side marked x, giving

your answer correct to one decimal place.

(b) Triangle DEF has a right angle at E. The length of side DE is 4.2 mm,

and the length of side DF is 6.6 mm. Find ∠EDF, giving your answer

correct to the nearest degree.

You may find it useful to sketch triangle DEF in your answer.

(c) (i) Figure 6 shows the triangle ABC, with all side lengths measured in

centimetres. Find ∠ABC, giving your answer correct to the nearest degree.

(ii) Find the area of triangle ABC in Figure 6, giving your answer correct to the nearest square centimetre.

(d) (i) Convert 98◦ to radians, leaving your answer in terms of π. [2]

(ii) Use your answer from part (d)(i) to find the area of a sector of a

circle of radius 2.8 centimetres and angle 98◦, giving your answer

correct to two significant figures.

**Question 5**

This question is based on your work on MU123 up to and including Unit 12.

You should use trigonometry, not scale drawings, to find your answers.

Laura is designing a computer game which simulates space exploration. She

controls a spaceship which she wishes to pilot from Earth to a neighbouring

star called Arcturus.

Arcturus lies 36.7 light-years away (in a straight line) from Earth. However,

Laura cannot pilot the spaceship in a straight line towards Arcturus, as a

comet lies in her path. Starting from Earth, she sets out in a straight line.

She chooses a direction which avoids the comet, but still gets her slightly

closer to Arcturus.

Once she has travelled in a straight line for 9.1 light-years, she stops the

spaceship. From her current position, she can see both Earth and Arcturus,

and the spaceship’s instruments show that the angle between her lines of

sight to Earth and Arcturus is 110◦

(a) Sketch a diagram of the situation, showing the points E for the position

of Earth, A for the position of Arcturus and S for the current position

of the spaceship. Mark in the angle and the lengths (in light-years) that

you are given. Join the three points with line segments to make the triangle EAS.

(b) Laura would like to calculate the distance between the spaceship’s

current position and Arcturus. She realises that in triangle EAS she

has two side lengths and an angle. She mistakenly concludes that she

can solve her problem with a single direct application of the Cosine

Rule, like in Example 9 in Subsection 2.2 of Unit 12.

Explain, as if directly to Laura, why she can’t use the Cosine Rule

directly in this way to solve her problem.

(c) (i) Use the Sine Rule to find the angle at A. Give your answer correct

to the nearest degree.

(ii) Use your answer to part (c)(i) to find the angle at E. Give your

answer correct to the nearest degree.

(iii) Use the Cosine Rule and your answer to part (c)(ii) to find the

distance between the spaceship and Arcturus. Give your answer

correct to two significant figures.

**MU123 Discovering Mathematics Assignment-The Open University UK**

**Question 6**

This question is based on your work on MU123 up to and including Unit 12.

In this question, you are asked to comment on a student’s attempt at

answering the question detailed below. Read the question and the student’s

incorrect attempt first, then answer the questions below.

**The Question**

Yusuf is a building manager for an office which has a room full of computer servers. The servers must be kept cool with a fan, or else they will overheat and malfunction. Yusuf buys a fan which oscillates to cover an angle of 52◦. The servers he would like to keep cool span the length of a wall which is 5 metres wide. Assume that the fan is placed centrally with respect to the room’s width.

Yusuf sketches the situation in Figure 7, where all lengths are measured in metres. The point F represents the fan’s position. Points A and B are the furthest reaches of the room’s width. The point C is the centre point of the room’s width. The length of F A is equal to the length of F B. Angle AF B is 52◦, so angle AF C is 26◦

Assuming the fan is strong enough, how far from the centre point C should the fan stand in order to be able to utilise the full 52◦ oscillation to cool the whole wall’s width?

The student’s incorrect attempt

We first calculate the angle at A, using the fact that all three angles

in triangle ACF add up to 180o . So we have

A = 180 (90 + 26) = 64.

So the angle A is 64o

Using the Sine Rule in triangle ABF, we have

So the length of BF is 5.7 m (to 1 d.p.).

The length of BF is the same as the length of AF, and triangle ACF is right-angled, so we have

Therefore, to 2 decimal places, the fan should be placed at least 6.34 m

away from the centre of the room.

(a) Look at the student’s answers for the lengths BF and CF. Using what

you know about right-angled triangles, and without performing any

further calculations, explain how you know that at least one of these

calculated lengths must be wrong.

(b) There are two places in the student’s attempt where a mistake has been

made. Identify these mistakes, and explain, as if directly to the student,

why, for each mistake, their working is incorrect.

(c) Do you think the student’s approach to solving the problem is the most

efficient method? Give a reason for your answer.

(d) Write out your own solution to the problem, explaining your working.

**Question 7**

In this question you are asked to reflect on how your studies so far on

MU123 fit in with your wider study interests and future study plans.

(a) Choose one topic from MU123 that you think is relevant to your wider

study interests. Explain why you think your chosen topic is particularly

interesting and/or relevant to your future study plans.

If you don’t find any of the topics from MU123 interesting or relevant to

your future study plans, then explain why.

(b) Do you feel confident in your chosen topic? If so, describe one step you

could take to maintain your confidence in years to come. If not, describe

one step you could take to help you work more confidently in that topic.

If you did not choose one specific topic in your answer to part (a),

choose one topic from MU123 in order to answer this part. For example,

you could choose a topic which you found particularly challenging.

**Question 8**

A score out of 5 marks for good mathematical communication over the entire TMA will be recorded under Question 8.

**MU123 Discovering Mathematics Assignment-The Open University UK**

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