Mathematics

Question 10

Use complex numbers to solve a parallel arrangement of impedances giving the answer in both Cartesian and polar form Use differential calculus to find the maximum/minimum for an engineering problem.

A sheet of metal is 340 mm x 225 mm and has four equal squares cut out at the corners so that the sides and edges can be turned up to form an open topped box shape.

Calculate:

(i) The lengths of the sides of the cut out squares for the volume of the box to be as big as possible.

(ii) The maximum volume of the box. Z Z1 Z2

Question 11

Using a graphical technique determine the single wave resulting from a combination of two waves of the same frequency and then verify the result using trig formulae.

A sheet of metal is 340 mm x 225 mm and has four equal squares cut out at the corners so that the sides and edges can be turned up to form an open topped box shape.

Calculate:

1. Using a vector or phasor diagram add the following waveforms together and present your answer in the form Vr = V sin(ωt+α).

Show all working including phasor diagrams.

Add the following together:

V1=10sin(ωt)

V2=20sin(ωt+)1. Verify your result by using trigonometric formulae.

Question 12

In calculating the capacity of absorption towers in Chemical Engineering it is necessary to evaluate certain definite integrals by approximate methods. Evaluate the following definite integrals by use of Simpson’s rule and the trapezoid rule using the given data (xi is an empirical function of x, yi is an empirical function of y):

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