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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering-Stage 2
(NFQ – Level 8)
Autumn 2006
Mathematics
(Time: 3 Hours)
Answer FIVE questions. All questions carry equal marks.
Examiners: Prof.M.Gilchrist Mr.J.E.Hegarty Mr. T O Leary
- (a) The current i in a LR circuit at ant instant t is found by solving the differential equation di +4i=50sin3t i(0)= dt By using a method of your own choice solve this differential equation. Express the steady state current as a single function of the form Rsin(ωt-α). Write down the maximum and minimum values of this function. Find the smallest positive values of t for which these extreme values hold. (8 marks)
(b) By using two different numerical methods with a step h=0.1 estimate the value of y at x=3.1 where dy (^) 2xy y(3) 1 dx =^ =^.^ (5 marks) (c) A closed rectangular box is to be constructed. It costs €10 per m^2 to construct the base and €6 per m^2 to construct the other sides. Find the dimensions of the box of largest volume that can be constructed for €192. Also find this maximum volume. (7 marks)
- (a) Show that the Taylor series expansion of f(x,y)=xln(2x-y) about the values x=2,y=
is gven by f(x,y)=4(x-2)-2(y-3)-2(x-2) 2 +3(x-2)(y-3)-(y-3) 2 +… (6 marks)
(b) Variables u and v are related to variables x and y by the formulae
u=ln 3y 2x
v=^ x^2 −^ y^2.
Find the partial derivatives of u and v with respect to x and y. (i) If T=f(u,v) is an arbitrary function in u and v write down the relationships between the partial derivatives of T with respect to x and y and those with respect to u and v. (ii) Estimate the value of v if the values of x and y were estimated to be 5 and 3 with maximum errors of 0.02 and 0.04, respectively. (8 marks)
(c) Find the maximum/minimum values of the function V=2x 3 +6xy+3y^2 +12y+4 (6 marks)
- (a) Find the Inverse Laplace transform of the expressions
(i) (^) s (^3) + 4s^242 + 4s (ii) (^) s(s (^2 120) + 2s+5) (11 marks)
(b) Using Laplace Transforms solve the differential equation 2 2
d x 6 dx 8x 80 x(0) x (0) 0 dt +^ dt +^ =^ =^ ′ =^ (4 marks)
(c) Find the zero and the poles of the transfer function L[f(t)]L[y] where
(^2) t (^2 )
d y 5 dy 24y 20 ydt f(t) y(0) y (0) 0 dt dt
+ + + ∫ = = ′ = (5 marks)
- In answering the following question you are required to use the Method of Undetermined
(ii) By calculating a double integral locate the centroid of this region. (10 marks)
(b) A semicircular region is described by R: x^2 +y^2 ≤ 4, x ≥ 0 (i) By evaluating an appropriate double integral find the second moment of area of the region about the x-axis
(ii) If V is the volume with a semicircular cross section described above and whose height is described by 0 ≤ z≤ 6 evaluate the triple integral
V
∫∫∫ xyzdV (10 marks)
- (a) A variate can only assume values between x=0 and x=3. Show that the function
p(x)= 181 (x^2 +2x).
is an acceptable p.d.f. Find the mean value of the distribution, P(1<x<2), and the median value of this distribution. This value is close to x=2.25. (7 marks)
(b) The mean diameter of bolts produced by a certain machine is 45 mm with an estimated standard deviation of 0.03 mm. What percentage of diameters are less than 45.05mm and what percentage lie between 44.95 and 44.98mm? If 0.2% are deemed to be undersize for being smaller than what some critical limit find this critical limit? If a sample of one hundred of such bolts were taken at random by using the Binomial Distribution calculate the probability that this sample contains at most two undersize bolts. (10 marks)
(c) The average demand to hire a particular item is 7 per week. If there are 5 working days in a week calculate the chances of three or more requests to hire a machine on a particular day? (3 marks)
f(x) f ′ (x)^ a=constant x n^ nxn- lnx x
e ax^ ae ax sinx cosx cosx -sinx tan-1^ (x) x 1
tan −^1 x a x 2 a +a 2
uv dx vdu dx u dv+
v
u
v^2 dx
udv dx v du−
f(x) ∫ f(x)dx a=constant
sinx -cosx cosx sinx e ax a
(^1) e ax
∫ UdV=^ UV−∫VdU
