ENM1600 Engineering Mathematics Assignment 3

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Question 1.

Find each of the following limits:                                                                                                [16 marks]

; .

Question 2. A rocket of mass m = 1000kg is travelling in a straight line for a short time. The distance in metres covered by the rocket during this time is described by the function

r(t) = t3− 3t2 + 6t

where t > 0 is the time in seconds.

[14 marks]

(a)    The kinetic energy E of the rocket is given by   , where v is the rocket’s speed. Find a function that describes the kinetic energy of the rocket.

(b)    Find the kinetic energy of the rocket at time t = 3 seconds.

(c)    What is the distance covered by the rocket by time t = 30 seconds?

(d)    Find the value of time t when the speed of the rocket is 120ms−1.

(e)    Find a function that describes the acceleration of the rocket.

(f)     Find the acceleration of the rocket at t = 3 seconds.

(g)    Find the time when the rocket’s acceleration is 27ms−2.

Question 3.

Find

[14 marks]

at the point  (−2,0),  if y3 = x3 + ex sin y + 8.

Question 4.

The work done by a variable force,  f(x),  is given by                                                         [16 marks]

b

W =         f (x) dx.

a

(a)    Find the indefinite integral of the force

F (x) = 9x 3 √x − 2x 5   + e −2x   +  11x

i.e. R f (x)dx.

(b)    Hence calculate the exact value of the work done by the force if a = 0 and b = 1 i.e. evaluate the integral

Question 5.

To help find the velocity of particles requires the evaluation of the indefinite [20 marks] integral of the acceleration function, a (t), i.e.

Z v =            a(t)dt.

Evaluate the following indefinite integrals:

Z

(a);

Z

(b)            (t2 + 1)sin9tdt.

Question 6.

City A and B are separated by a 2km wide river and are located as shown          [20 marks] in Figure 1 (not drawn to scale). A road is to be built between city A to B that crosses a bridge straight across the river. Where should the bridge be built (i.e. what is the value of x) so that the road between city A and B is as short as possible? What is the minimum length of the road?

A

Figure 1: Proposed road between A and B.

Total: 100 marks