Question: The understanding of more advanced mathematics is important within an engineering curriculum to support and broaden abilities within the applied subjects at the core of all engineering programmes. Students are introduced to additional topics that will be relevant to them as they progress to the next level of their studies,

advancing their knowledge of the underpinning mathematics gained in Unit 2: Engineering Maths.

The unit will prepare students to analyse and model engineering situations using mathematical techniques. Among the topics included in this unit are - number theory, complex numbers, matrix theory, linear equations, numerical integration, numerical differentiation, and graphical representations of curves for estimation within an engineering context. Finally, students will expand their knowledge of calculus to discover how to model and solve engineering problems using first and second order differential equations.

On successful completion of this unit students will be able to use applications of number theory in practical engineering situations, solve systems of linear equations relevant to engineering applications using matrix methods, approximate solutions of contextualised examples with graphical and numerical methods, and review models of engineering systems using ordinary differential equations.

Unique Numbers

You will be given a set of numbers - these are unique values for your assessment. (*) denotes student number

Note: due to student number system your answer may not be realistic. Adopt an accurate real-world approach to using your number correctly.

Task 1

The latest product designed by your department requires testing to ensure accuracy, reliability and functionality. Undesirable energy loss and waste has been identified in a circuit upgrade being produced. The power being dissipated, P in the resistor R is given by- P = V2R

Where V is the voltage across the resistor, plot on the same axis the graphs of P against V for 0≤ V ≤ 5 and

a) R=(5+∗)Ω

b) R=(10+∗)Ω

c) R=(20+∗)Ω

What happens to the power dissipated for an increase in resistance?

Task 2

To support a colleague`s work, with fluid dynamics system flow problems, you are helping them to investigate their findings. The velocity flow, v, of the liquid, along a channel satisfies the following -

v3-6v2-348v+3112=0

Given that there is a root of this equation between v=10 and v=11. Find this root correct to three decimal places by the Bisection method.

Task 3

Your employer has proposed a meeting to discuss a large contract that has been put out to tender internationally. The opportunity is to secure large development projects based on steelwork design in earthquake prone countries. You are to devise a test procedure regarding beam oscillations. To test you are looking at the critical speeds of oscillation y of a loaded beam and know that the oscillation is given by the equation- y3-3.250y2+y-0.063=0

You have decided to use the Newton Raphson method to numerically calculate the value of y (which you know to be approximately equal to 3.0). You have decided that 3 decimal places are an accurate approximation.

Task 4

Your present position is working at a pharmaceutical company. Within the research and testing facilities an issue has occurred regarding samples dissolving incorrectly after a new production system has been tested. A problem has been identified w regarding issues with force, F, acting on a particle with time, t, according to the table below:

t (s) 0 0.5 1 1.5 2 2.5 3

F (N) (*+3.2) (*+5.6) (*+7) (*+7.7) (*+8.4) (*+9.9) (*+11.6)

The impulse of this force is given by

∫F.dt30

Find an approximate value for the impulse using the trapezoidal rule

Task 5

You are asked to check some data from a logistics program that is being designed to measure vehicle dynamics to improve reliability and efficient by a productive reduction in travel. The test vehicle starts from rest and its velocity is measured every second for 6 seconds with the following results-

Time t (s) 0 1 2 3 4 5 6

Velocity v (m/s) 0 (1.2 +*) (2.4+*) (3.7+*) (5.2+*) (6+*) (9.2+*)

Using Simpsons rule, calculate the distance travelled in 6 seconds (this is the area under the velocity time graph)

Task 6

Further to your previous oscillation test results, the I beams you manufacture will require testing with regard to the quality from different suppliers. The equation for force, on a beam of length *m is given by. F=∫ex1+√x dx∗0

Where x is distance.

Determine the force on the beam of length * m

Task 7

A team visit has been arranged to the NASA HQ as a new policy in team building at your workplace. As an afternoon team challenge you are asked to measure the performance of a rocket based on the following information -

The height, s of a rocket in the first * seconds is given by:

s=[5×103∫ln(100100-4t)t0dt]-180

Determine the height of the rocket after t= * seconds

Task 8

You have been asked by a regional manager for advice on an alternative solution to a problem without incorporating calculus due to difficulties that they have encountered. Prepare a critically evaluation of the use of numerical estimation methods for your manager, commenting on their applicability and the accuracy of the methods.