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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 2 Examinations 2007/
Module Title: Technological Mathematics 1
Module Code: MATH 6014
School: School of Mechanical & Process Engineering
Programme Title: Bachelor of Engineering in Mechanical Engineering – Stage 1
Programme Code: EMECH_7_Y
External Examiner(s): Dr. P. Robinson Internal Examiner(s): Ms. J. English Dr. S. O’Rourke
Instructions: Answer Question 1 (worth 40 marks) And two other questions (worth 30 marks each)
Duration: 2 HOURS
Sitting: Summer 2008
Requirements for this examination:Graph Paper and Log Tables are available
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the correct examination paper. If in doubt please contact an Invigilator.
Q1. (i) Given the formula
t cgh^2 L = + t + h Find the value of L when t = 9.3 × 10 −^3 , c = 7.1, g = 3.6 × 10 −^1 and h = 6.5 × 10 −^1. (ii) Solve for x : log ( 2 x + 2 ) + log ( 2 x − 2 ) = 3
(iii) Simplify (^2) x^3 +^5 − x^4 − 1
(iv) Determine if the following equation has real roots 3 x^2 − 5 x = − 11 (v) Use the data in the following table to find values for s and n in the relationship (^) N^ S 2^ = aN + b.
S 13 55 N 2.4 6. (vi) Write the relationship ln P = − kt + ln A in exponential form. Find the value of P when k= 0.004, t = 100 and A = 70. (vii) Consider the function y(t)= 7 cos(15 π t + 0.68),where t is measured in seconds and y is measured in metres. Determine the amplitude, periodic time, frequency and phase angle of y(t) and states whether the phase angle lags or leads y(t). (viii) In a triangle ABC, side a = 6.2cm, c = 4.8 cm and B = 72 , determine side^0 b. (5 marks each)
Q4 (a) Express the following in linear form, indicating what you would plot along each axis and what each constant represents.
(i) P = (^) B + T kT B and k are constants.
(ii) V = kTn k and n are constants. (10 marks) (b) The law Q = aHb^ is thought to apply to the following data where a and b are constants. H 4 8 12 20 Q 20 28.3 34.6 44.
Verify this law applies by plotting a suitable linear graph. Solve for constants a and b. (20 marks) Q5. (a) Prove the identity 1 cos sin sin 1 cos
x x x x
(8 marks) (b) The value of the voltage in a circuit at any time t seconds is given by: v t ( ) = 210sin(20 π t +0.38) volts Determine (i) The value of v(t) when t = 0 (ii) The value of v(t) when t = 20ms (iii) The time when v(t) is a maximum. (iv) The time when v(t) reaches 100 volts. (14 marks) (c) Find the values of B in the range 0 o^ ≤ B ≤ 360 o which satisfy the equation cos B = -0.8. (8 marks)
