# SUMMATIVE ASSIGNMENT – Mathematics for Science

IMPORTANT INFORMATION

• You must show your working – marks are not awarded if only the answer is given.
• Formulae are generally not included – you will need to consult the module materials or other reliable sources to access these.
• You should answer all questions – 100 marks are available.
• Remember to include appropriate units with your answers. If you are giving an answer in SI units, you should convert to the appropriate unit(s) if necessary before you calculate.
• Submit your completed assignment on the module moodle site as a WORD (not PDF) document by 12th January 2024 (DDS and EC up to 5 days later).

1

Electric power is widely used in industrial, commercial and consumer applications. The latter include laboratory equipment for example water baths, spectrophotometers, and chromatographs.

If you have 17.3 kA and 5.5 MV, what is the power? Give the appropriate unit.

(3 marks)

2

Oil immersion objective lenses are used in light microscopy, and it is important to know the refractive index of the oil to obtain a clear image. The refractive index (n) of an oil changes with temperature (T) according to the following formula: –

n = 1.5811 – 0.000438 x T, where T is the temperature in degrees Celsius.

1. Calculate the refractive index of the oil at a room temperature of 210C.                                                                                                      (2 marks)
2. At what temperature will the refractive index of the oil be 1.5556?

(3 marks)

3

In a synthetic chemical reaction, the theoretical yield of the product is based on the mass of the reactants used and takes account of the stoichiometry of the chemical reaction. Calculate the following percentage yield of a reaction if 4.31 grams of product was obtained, when the theoretical yield was 6.57 grams.                                                                                                                                            (2 marks)

4

Disposable cone-shaped receptacles are often used in conjunction with drink dispensers for example water coolers. It is therefore necessary to know the capacity of the receptacles.

1. Calculate the volume of a cone where radius = 4.5 cm and height = 8.7 cm                                                                                                (2 marks)
• Make height (h) the subject of the formula, and calculate the height of a cone whose volume is 353 cm3 and whose radius is 4.8 cm.                      (2 marks)
• Pythagoras’ theorem states that a2 + b2 = c2 where a and b are the lengths of the two shorter sides (h and r in the diagram) of a right-angled triangle and c is the length of the longest side (the hypotenuse, shown as L in the diagram).

Calculate L using the value of the height calculated in (b) and radius = 4.8 cm

(2 marks)

5

Astronomers are able to observe planets within our solar system due to reflected light from the sun. The speed of light is 2.99 x 108 metres per second. Light from the sun takes 6 hours, 23 minutes, and 7 seconds to reach Pluto.

1. How many seconds are there in 6 hours, 23 minutes, and 7 seconds?

(1 mark)

• How far has the light travelled in that time? Give your answer in metres and in standard form                                                                                (2 marks)
• The mass of Mars is 6.390×1023 kg, and that of Jupiter is 1.898×1027 kg.

How many timesheavier is Jupiter than Mars? Give your answer in standard form.                                                                                        (3 marks)

6

The density of aluminium is relatively low in comparison to many other metals. This means that large volumes of aluminium are comparatively light. Consequently, aluminium is a major component of an aircraft fuselage.

The density of aluminium is 2.3 x 103 kg/m3.

a) Calculate the mass of an aluminium block 5cm x 5cm x 3cm.            (2 marks)

b) Calculate the mass of a solid aluminium cylinder 10cm long and radius = 1cm.

(2 marks)

(c) Calculate the volume of 0.2kg of aluminum.                                     (2 marks)

7

In a case of a driver involved in a car traffic accident who is suspected of driving under the influence of alcohol, it might be a requirement to know the rate of alcohol degradation within the body. The amount of alcohol in blood reduces by 0.135 g/l/hr. (In every hour, the alcohol by volume (ABV) decreases by 0.135 g/l).

The driver is found to have an ABV of 1.60 g/l.

a) How many hours will it take for his or her ABV to fall to the legal limit of 0.8 g/l?                                                                                                         (3 marks)

b) If the ABV was 2.55 g/l at 3.00am, at what time will the ABV be at the legal limit?                                                                                                      (3 marks)

8

The Beer-Lambert law correlates the concentration of a solution with its absorbance of light. This allows the construction of a calibration curve to determine the concentration of samples of the solution.

1. Calculate the molar extinction coefficient (ε) for a chemical whose absorbance = 0.17 at a concentration of 0.00055M, with a path length of 1 cm.                                                                                                                                                                                                                                    (3 marks)
• Using the calculated value of ε, give the concentration of this chemical at an absorbance of 0.24, path length = 1 cm                                            (3 marks)

9

Ketoacidosis, leading to low blood pH, is a possible indication of diabetes mellitus whereby increased amounts of acidic ketone bodies are produced.

Calculate the pH of solutions where [H+] =

a) 7.0 x 10-2 moles/litre                                                                  (2 marks)

b) 2.6 x 10-5 moles/litre                                                                  (2 marks)

c) 5.3 x 10-11 moles/litre                                                                 (2 marks)

10

The pH of urine can provide clinical information for example as part of an assessment of kidney function. A solution has a pH of 4.755. How many moles of hydrogen ions are dissolved in 0.350 dm3 of it.

11

Bacteria are a major component of intestinal flora and play a key role in fine-tuning the immune system. It has been estimated that on average, the human body contains 92 grams of bacteria. The mass of a single bacterial cell is 8.0 x 10-16 grams. How many individual bacterial cells are there? Give your answer in standard form.                                                                                                         (2 marks)

12

Solve the following simultaneous equations.

6x + 2y = 18

3x + 3y = 21                                                                                            (2 marks)

6x – y = 21

4x + 2y = 38                                                                                            (2 marks)

13

Argon-ion lasers are used as an excitation source in confocal microscopy. The radius of an argon atom is 7.1 x 10-10 decimetres. The argon atom is spherical.

1. Calculate the volume of 1 (spherical) argon atom in cubic decimetres (dm3).                                                                                           (2 marks)

b) There are 6.022 x 1023 atoms (or molecules) in one mole. Calculate the volume of all the atoms in one mole of argon. Give your answer in dm3.

(2 marks)

14

The reaction times of an individual is often conducted as part of a neurological assessment, and the reproducibility of the reaction times could be significant to the investigation. The standard deviation of a data set provides information in relation to the spread of the values within the data set. An experiment was conducted to measure a set of reaction times of an individual.

The following eight reaction times were recorded (in seconds).

Reaction Times = 0.17, 0.19, 0.24, 0.21, 0.24, 0.21, 0.20, 0.24

1. Calculate the mean, mode and median                                    (1 mark each)
• Calculate the Variance                                                                      (2 marks)
• Calculate the Standard Deviation                                                       (2 marks)

15

Ballistics is the branch of scientific investigation involving the use of firearms. It is important to determine the velocity of the bullet to gain information of the distance travelled by the bullet (trajectory).

A bullet fired from a gun has a velocity of 600 km/hr and a mass of 15 grams.

1. Convert the velocity into metres/second.                                          (2 marks)
• Convert the mass into kilograms.                                                     (1 mark)
• Calculate the kinetic energy (KE) of the bullet in joules.                     (2 marks)

The bullet was fired vertically. It will go up until all the KE is converted to Potential Energy (PE). At this point, the bullet stops, and starts to fall back to earth.

Calculate how high the bullet goes before it stops, assuming all the KE is converted to PE.                                                                                                (3 marks)

16

The correlation of glucose concentration and density shows a linear relationship. A hydrometer is used to determine density (specific gravity) and has applications where sugar concentrations need to be monitored for example in wine making, brewing and ice-cream production.

An experiment to correlate the density of glucose solutions at a range of concentrations was conducted. The results are shown in graphical form below.

Use the equation to calculate the density of the glucose solution at a concentration of 15.82 % w/v.                                                                                                (3 marks)

The equation is shown in the form y = mx + c. What are m and c?    (1 mark each)

17

In a chemical reaction, the surface area of the reactants is a major determinant of the reaction rate. The greater the surface area, the larger the area available for the interaction of the reactants. The following question concerns the surface area of a sphere.

Calculate the cost of electroplating the surface of a metal sphere of 30mm diameter at a cost of £2.25p per cm2                                                                                                    (3 marks)

18

Grass contains the light-harvesting pigment called chlorophyll, which is the key compound for the process of photosynthesis. Jack wants to buy enough grass seed for his garden. Jack’s garden is in the shape of an isosceles trapezium (the sloping sides are equal in length). Each box of grass seed covers 10m2. The angle shown measures 78o.

How many boxes of grass seed will Jack need to buy?                    (5 marks)

19

Potassium chloride is a major component of lethal injections as it quickly causes cardiac arrest. A researcher wants to make 300 cm3 of a potassium chloride (KCl) solution at a concentration of 0.15M.

(atomic mass of potassium = 39.0, atomic mass of chlorine = 35.5)

Calculate the mass of KCl that needs to be dissolved in 300 cm3 of water.

(3 marks)

200 cm3 of this solution was used and the researcher took the remaining 100 cm3 to produce 100 cm3 of KCl solution at a concentration 0.25M. What is the mass of KCl that needs to be added to the 100 cm3 portion to create the 0.25M solution?

(3 marks)

20

The relationship between distance and velocity is given by s = ut + 0.5at2

Where s = distance, u = velocity, a = acceleration due to gravity and t = time

a) Make u the subject of the formula                                                         (2 marks)

b) Calculate u if s = 90.4metres, a = 9.8m/s2 and t = 4 seconds                  (2 marks)

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