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

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