Matths assessment 4 - math work and equations PDF

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Question 1 Getting started Europa of the Jovian system is from the surface just a moon of ice and dust, but under the surface the satellite reveals to us a global subterranean ocean within which human colonization may depend upon. But moving to other moons is no day trip, the distances within simply our own cosmic neighbourhood are staggering, and to transport life across the solar system we need to understand these distances in relation to the only thing vast enough to our minds to appreciate the work required in space colonization: the human lifetime. Europa can be anywhere from 33 to 54 light minutes from earth, and the average human life expectancy is 71.5 years (Central Intelligence Agency, 2017). The space shuttle within which humans might fly can travel at 9200ms-1 and the speed of light is given as 3x108ms-1. Two possible logistical methods for getting people to Europa to be explored will be the possibility of seeding young children to arrive fully matured, and to send breeding pairs to conceive in transit. Both methods will weigh on assumption of thought hypothesis but due to lack of existing data on the subject these are the methods most readily available to this examination.

Written response: Problem solving Calculations 

All workings have been rounded 3 decimal places.

Firstly, the speed the of light per minute will be worked out. (3 x 108ms-1) x 6 x 101s = 1.8 x 1010 m 33 x 1.8 x 1010m = 5.94 x 1011 m[min] 54 x 1.8 x 1010 m= 9.72 x 1011m[max]

5.94 x 1011 m / 9200 = 64565217.39s 64565217.390s / 3600 = 17934.782h 17934.782s / 24 / 365 = 2.047y Moving at 9200m/s as the crow flies a space shuttle would take 2.047 years with minimum distance. 9.72 x 1011m / 9200 = 105652173.900 105652173.900 / 3600 = 29347.826h 29347.826 / 24 / 365 = 3.350y Moving at 9200m/s as the crow flies a space shuttle would take 3.35 years at a maximum distance. -

Minimum age on arrival should be 10 years old

-

Maximum age 45 years old

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Written response: Problem solving Conclusion The conversion of values serves as a useful tool to find an appreciable perspective on the distances and time required for transit. Starting with standard measure of light minutes, the distances are converted to metres and the time is converted to years, to put the duration of flight into the perspective of a human lifetime . Limitation arises when considering that spaceflight is not a linear process as described above. The required information to accurately describe the path and duration of a factual transit is not given, however the precision of this fictitious journey is still true. Alternatively, calculation could be made by observation, as several artificial satellites have been sent to and passed by Europa, and such the time and distance could be found by their journeys to the Jovian moon. The results would be imprecise for a future mission due to their orbital transits, but the information would be more accurate than a purely linear trajectory. The calculations for such a flight are, however, beyond the scope of this question. The application of such calculations is not yet readily of use to any industry save for national space agencies, but the future probability of manned missions to the outer solar system becomes more likely each passing year. It is possible that in a far future where such a colony exists, the technology to travel a linear path, and require the former method, will be possible.

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Written response: Problem solving Question 2 Getting started In December, two ice creameries are looking to sell their businesses, and are analysing their sales from over the year between June and December. Both salesmen understand the ice cream sales are dependent on the weather, with an understandable spike in sales owing to a proportionate spike in temperature. The real-estate has collected the data from both stores, one in a shopping complex and one by a beach, and is comparing and contrasting to determine the value of both businesses. To find the true relationship between weather, location and sales, the data is best represented in a graph which can be algebraically manipulated to determine the strength of each variable’s impact on one another. Calculations Table 1. Ice-cream sales versus temperature Ice Cream Sales vs Temperature Temperature °C

Sales in Shopping Centre

Sales at Beach Shop

11.9 14.2 15.2 16.4 17.2 18.1 18.5 19.4 22.1 22.6 23.4 25.1 30.1 33.0 34.0 37.0

$185 $215 $332 $325 $408 $421 $406 $412 $522 $445 $544 $614 $700 $725 $780 $801

$165 $180 $200 $200 $251 $251 $300 $395 $400 $425 $451 $500 $600 $650 $800 $1,050

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Written response: Problem solving

Ice cream sales Total sales in $AUD

$1,200 $1,000 $800

f(x) == 24.63 31.74 xx −− 61.73 284.35 f(x)

$600 $400 $200 $0

10

15

20

25

30

35

Temperature in Degree C Sales in Shopping Centre Sales at Beach Shop

Linear (Sales in Shopping Centre) Linear (Sales at Beach Shop)

Figure 1. Ice cream sales It is shown through the linear equation y=mx+b that the line of best fit for the value of

the shopping centre can be shown with the equation

x1 − x2 y1 − y2

. Inserting the values of

(18,400) for (x,y)1 and (33,750) for (x,y)2 gives us a final gradient m=23.33. This is represented in dotted blue as y=24.63x + (-61.726) (Fig.1) It is shown through the linear equation y=mx+b that the line of best fit for the value of

the beach shop can be shown with the equation

x1 − x2 . Inserting the values of y1 − y2

(15,200) for (x,y)1 and (34,800) for (x,y)2 gives us a final gradient m=31.57. This is shown in dotted orange as y= 31.736x + (-284.35). (Fig. 1).

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40

Written response: Problem solving

Conclusion The method of substituting values into known equations to find analytical results clearly shows that sales are greater in the shopping centre, however on warmer days the beach shop is clearly the more popular of the two establishments. However as not every day is conducive to recreation at the beach, the shopping centre is a more valuable store. The assumption that sales would increase on hotter days is reflected in the data, and a further assumption could be made that factors beyond sales may impact the value of both stores, such as location or public sentiment which are not reflected in this data set. The substitution method is the simplest way to extrapolate the line of best fit from a scatterplot such as given. Where this graphed by hand rather than digitally, a rough line of best fit might have been found by eye and drawn onto the field, however the digital method is more accurate and precise due both to its reliance on purer mathematics and the reliable nature of digital calculation. The real-world applications are clear in that small businesses are bought and sold daily and a fast and accurate method of appreciation is not only necessary but a cornerstone of modern real estate.

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Written response: Problem solving

Question 3 Getting started The 2014-2015 National Health Survey (NHS) is analysing data collected on age demographics in the health system, and the frequency of attendance to sessions with a general practitioner (GP) in each age demographic bracket. The aim of the survey is to assess the required support in rural and regional areas, and to present the data in a fluid and concise manner to the regional health community group. Data has been collected from all ages, and is divided between GP’s, dentistry, and specialists. To support the case for an expansion in regional GP’s, the most effective methods would to graph and statistically analyse the given data. Calculations Table 2. Data collection Age in years

GP

Specialist

0–14 15–24 25–34 35–44 45–54 55–64 65–74

3,574.70 2,352.50 2,860.70 2,588.40 2,572.50 2,453.40 1,842.50

927.3 760.4 947.7 1,025.20 1,076.50 1,275.30 1,077.10

Dentist 2,101.20 1,402.40 1,328.20 1,474.80 1,444.20 1,446.10 971.2

75 years and over

1,322.80

832.3

581

Mean

2445.94

989.1

1343.64

Table 3. Data aggregation Age in years 0–14 15–24 25–34 35–44 6|Page

Healthcare Population Total 6603.2 4515.3 5,136.60 5,088.40

Written response: Problem solving 45–54 55–64 65–74

5,093.20 5,174.80 3,890.80

75 years and over

2,736.10

Specific Healthcare access by age (x1000) 75 years and over 65–74 55–64 45–54 35–44 25–34 15–24 0–14 0.00

1,000.00

2,000.00

3,000.00 GP

4,000.00 Specialist

5,000.00

6,000.00

7,000.00

6000

7000

Dentist

Figure 2. Specific Healthcare access by age

Healthcare access by age (x1000) 75 years and over 65–74 55–64 45–54 35–44 25–34 15–24 0–14 0

1000

2000

Figure 3. Healthcare access by age

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3000

4000

5000

Written response: Problem solving

Conclusion The given data, though not enough to form a conclusive argument for or against the increase of funding in regional healthcare, creates a strong narrative of the national state of healthcare. Observed is a clear change over age groups in which healthcare services are accessed, which gives an insight into the needs across age groups. A clear dip in healthcare access is seen in youth (15-24), whilst another, steady dip towards the end of the age spectrum, indicates the aging population coming to death, and thus no further need for healthcare. Also seen is the flux of GP’s contrasted by specialists, and dentists and specialists. Where the rate of GP and dentistry access declines as age increases, the rate of specialist access increases with age. Data is only collected between the years of 2014 and 2015, and thus is latitudinally limited, though the snapshot of the state of healthcare access has its uses. The lack of specification of specialisations makes the given data very broad and very shallow in term of usefulness, and by singling out dentistry deduction could be conflated about the nature of dentistry in the Australian health service and its relationship with the rest of the specialization community. The lack of geographical demographics, however, is the main reason this dataset is mostly useless to the regional health community group. It could be argued that the regional health centres hold a lack of staff proportionate to the age demographic, such as too few specialists to and aging population or a need for youth education and encouragement to seek healthcare access, however any argument of this kind would be unsubstantiated by the given information, and thus invalid in an objective statement for an increase in regional health funding and staffing. The alternative is to collaborate with a census of the regional population, and cross analyse the needs of the community with the existing establishment of regional GP’s. This is, however, beyond the scope of given information.

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Written response: Problem solving

Reference Central intelligence agency 2017 https://www.cia.gov/library/publications/the-worldfactbook/fields/2102.html

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