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Cork Institute of Technology
Bachelor of Engineering (Honours) in Structural Engineering - Stage 2
(CSTRU_8_Y2)
Autumn 2008
Mathematics
(Time: 3 Hours)
Answer FIVE questions. All questions carry equal marks. Statistical tables are available.
Examiners: Mr. P.Anthony Prof. P.O Donovan Mr. T O Leary
- (a) Solve the differential equation
dy +2y=12x y(0)= dx
By using two numerical method with a step of 0.1 estimate the value of y at x=0.1. (9 marks)
(b) By evaluating an appropriate double integral find the first moment of area about the x-axis for the region in the first quadrant bounded by the parabola y=x^2 and the line y=2x. Sum vertically and sum horizontally. (6 marks)
(c) State Rolle’s Theorem. Show that the function f(x)=2x 3 -6x^2 +4x satisfies the criteria of the theorem over the interval [0,1]. Find the critical value that satisfies the conclusion of the theorem. (5 marks)
- (a) Variables α and z are related to variables x and y by the formulae
α=f(x,y) =arctan ^ 3y2x
z= x -4y^2
(i) Find a Taylor Series expansion of the function f(x,y) about the values x=2, y=1. The series is to contain terms obtained from second order partial derivatives.
(ii) If T=f(z,α) is an arbitrary function of z and α write down the relationships between the partial derivatives of T with respect to x and y and those with respect to z and (^) α. Express
x T^ y T x y
∂^ + ^ ∂
∂ ^ ∂
in its simplest form. (iii) Estimate the value of z where the values of x and y were estimated to be 5 and 2 with maximum errors of 0.06 and 0.03, respectively. (14 marks)
(b) An open rectangular box with a square base is to contain 9 m 3 of liquid. In order to reduce heat loss the base is insulated at a cost of €8 per m 2 and the sides at a cost of €12 per m 2. By using a Lagrangian Multiplier and by using a second method find the dimensions of the box so that the cost is at a minimum. Find this minimum cost. (6 marks)
- (a) By completing the square and by using partial fractions find the Inverse Laplace Transform of the expression
2
2s+ s -6s+
Note: coshA e^2 e sinhA e^2 e
A A A A = +^ = −
− − (6 marks)
(b) By using Laplace Transforms solve the differential equations
(i)
2 2
d y 6 dy+8y = 24 y(0)=y (0)= dt dt
(ii)
(^2) 2t 2
d y (^) 4y = 32e y(0)=y (0)= dt
(iii)
2 2
d y +4 y =16t y(0)=y (0)= dt
′ (^) (14 marks)
(b) If C is the perimeter of the triangular region with vertices (-1,0), (1,0) and (0,1) evaluate the line integral 2 2 C
Ñ∫12x dx+12y dy.
With the aid of a double integral find the second moment of area of this triangle about the y-axis. (9 marks)
(c) If V is the volume with a constant cross section that is described by x 2 y^2 9 +^4 ≤^1 0 ≤^ z^ ≤^2 evaluate the triple integral 2 2 V
∫∫∫ (8x +18y )dV (4 marks)
- (a) Find the equation of the line passing from A(1,1,1) to B(2,2,0). If A =2z i +3y j -2z k and C the line passing from A to B evaluate the line integral
C
∫ A .d^ r^ (4 marks)
(b) Using the Method of Variation of parameters find the general solution of the differential equation 2 x 2 2
d y 2 dy (^) y 10e dx −^ dx +^ =^ x (7 marks)
(c) Consider the set of simultaneous equations dx (^) 4x y x(0) 3 dt dy (^) x 2y y(0) 2 dt
(i) Find the general solution for x and for y. (ii) By using a different method solve for x. (9 marks)
- (a) The weights of blocks are assumed to be Normally distributed with a mean value of 3.05kg and with a standard deviation of 0.03kg. (i) Calculate the percentage of bricks that weigh between 3.00kg and 3.12kg. (ii) If 99.8% of weights are greater than W find the value of W. (4 marks) (b) Samples of 50 items are taken from the output of a machine and for 8 such samples the number of defective items were counted: 2, 0, 1, 1, 2, 1, 1, 2. Calculate the mean defective rate. What are the chances of a sample of 150 items containing three or more defectives? Use both the Binomial and Poisson Distributions (7 marks) (c) A variate x can only assume values between 0 and 1 and its probability density is given by
p(x)= A(2x + 6x^2 ).
Find the value of A, the mean value of the distribution, the expected value of x. Also find correct to two places of decimal the median value of the distribution. This value is close to x=0.. (9 marks)
f(x) f ′ (x)^ a=constant x n^ nxn- lnx x
e ax^ ae ax sinx cosx cosx -sinx tan-1^ (x) 2
1+x tan 1 x a
− ^
a +x uv dx vdu dx u dv+
Note: coshA e^2 e sinhA e^2 e
A A A A = +^ = −
− −
