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ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 1 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 1 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 1 of 4

ENM1600 Engineering Mathematics, S1–2021

Assignment 3

Value: 10%. Due Date: Tuesday 25 May 2021.

Submit your assignment electronically as one (1) PDF file via the Assignment 3 submission

link available on study desk before the deadline. You may edit/change your assignment whilst

in Draft status and it is before the deadline.

Please note: Once you have chosen “Submit assignment”, you will not be able to

make any changes to your submission.

The submitted version will be marked so please check your assignment carefully before submit-

ting.

Hand-written work is more than welcome, provided you are neat and legible. Do not waste time

type-setting and struggling with symbols. Rather show that you can use correct notation by

hand and submit a scanned copy of your assignment. You may also type-set your answers if

your software offers quality notation if you wish.

Any requests for extensions should be made prior to the due date and must be made via the form,

which can be found at https://usqassist.custhelp.com/app/forms/deferred_assessment.

Requests for an extension should be supported by documentary evidence.

An Assignment submitted after the deadline without an approved extension of time

will be penalised. The penalty for late submission is a reduction by 5% of the maximum

Assignment Mark, for each University Business Day or part day that the Assignment is late.

An Assignment submitted more than ten University Business Days after the deadline will have

a Mark of zero recorded for that Assignment.

You must have completed the Academic Integrity training course before you submit this assign-

ment.

We expect a high standard of communication. Look at the worked examples in your texts, and

the sample solutions in Tutorial Worked Examples to see the level you should aim at. Up to

15% of the marks may be deducted for poor language and notation.

Please note: No Assignments for Assessment purposes will be accepted after as-

signment solutions have been released.

ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 2 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 2 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 2 of 4

QUESTION 1 (16 marks)

Find each of the following limits:

(a) lim

t→−7

5t2 + 24t− 77

4t2 + 35t+ 49

; (8 marks)

(b) lim

t→∞

420t3 − 7t4 − 28t9 + 18

(t6)

√

23

t

− 13t3 + 676t6 + t5

; (8 marks)

QUESTION 2 (14 marks)

A rocket is travelling in a straight line for a minute. The distance in metres covered by the rocket

during this time is described by the function

s(t) = 2t2 + 10t+ 128− 40 ln

(

t2 + 4

)

where t ≥ 0 and time is given in seconds.

(a) Find a function that describes the speed of the rocket. (3 marks)

(b) What is the speed of the rocket at the time t = 6 seconds? (1 mark)

(c) Find all values of time t (if any) when the speed of the rocket is 10 m s−1. (3 marks)

(d) Find a function that describes the acceleration of the rocket. (3 marks)

(e) Find the acceleration of the rocket at t = 4 seconds. (1 mark)

(f) Find all values of time t (if any) when the rocket’s acceleration is 4 m s−2. (3 marks)

QUESTION 3 (14 marks)

The position of a drone at time t (in minutes) is given by the parametric function

x = 5 tan−1 2t+ 7 e−7(t−1) sin

(

t2 − 3t+ 2

)

y = 6 ln (t XXXXXXXXXXe−7(t−1) cos

(

t2 − 3t+ 2

)

where both x and y are measured in metres.

(a) Find an expression for

dy

dx

in terms of t. (11 marks)

(b) Using part (a) evaluate the derivative when t = 1. (3 marks)

ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 3 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 3 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 3 of 4

QUESTION 4 (20 marks)

Mr Clarkson finds that the time taken to mow his lawn increases the longer he delays mowing. From

past experience he finds the time taken (in hours) to mow the lawn once, if he has not mown for t

weeks, is approximated by the function

C(t) =

4t

(

t2 − 5t+ 7

)

3t2 − 16t+ 24

.

(a) Find an expression for the total time spent mowing per year assuming he mows regularly every

t weeks.

You may assume there are 52 weeks in a year and the time taken to do the actual mowing is

incorporated into the value of t. (4 marks)

(b) If Mr Clarkson is to mow the lawn on a regular basis (every t weeks) determine, using calculus,

the optimal length of time should he delay mowing to minimise the total time per year he has

to mow.

Check your value of t by substitution into the derivative and verify that you have found the

minimum by using an appropriate calculus test.

What is the minimum total time spent mowing per year? (16 marks)

QUESTION 5 (16 marks)

The speed of a car at time t (in seconds) is given by a piecewise function V (t) (in m/s) as shown

below. Determine the total distance, d, travelled by the car from t = 0 seconds to t = 34 seconds.

t (s)

V (m/s)

V (t) =

7 t2

√

t

81

, if 0 ≤ t ≤ 9,

21− 11

π

cos

(

πt

2

)

, if 9 ≤ t ≤ 19,

21 exp

(

−(t− 19)

3

)

, if 19 ≤ t ≤ 34.

XXXXXXXXXX 32

4

8

12

16

20

24

Note: the exponential function, ex, can be written as exp (x).

ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 4 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 4 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 4 of 4

QUESTION 6 (20 marks)

To help find the velocity of particles requires the evaluation of the indefinite integral of the acceleration

function, a(t), i.e.

v =

∫

a(t) dt.

Evaluate the following indefinite integrals.

Check your value for each integral by differentiating your answer.

(a)

∫ (

27t2 − 12t− 9

√

t

)

ln t dt; (8 marks)

(b)

∫

7t2

(

27 cos 3t− 8e−2t

)

dt; (12 marks)

End of Assignment XXXXXXXXXXmarks Total)

ENM1600 Engineering Mathematics, S1–2021

Assignment 3

Value: 10%. Due Date: Tuesday 25 May 2021.

Submit your assignment electronically as one (1) PDF file via the Assignment 3 submission

link available on study desk before the deadline. You may edit/change your assignment whilst

in Draft status and it is before the deadline.

Please note: Once you have chosen “Submit assignment”, you will not be able to

make any changes to your submission.

The submitted version will be marked so please check your assignment carefully before submit-

ting.

Hand-written work is more than welcome, provided you are neat and legible. Do not waste time

type-setting and struggling with symbols. Rather show that you can use correct notation by

hand and submit a scanned copy of your assignment. You may also type-set your answers if

your software offers quality notation if you wish.

Any requests for extensions should be made prior to the due date and must be made via the form,

which can be found at https://usqassist.custhelp.com/app/forms/deferred_assessment.

Requests for an extension should be supported by documentary evidence.

An Assignment submitted after the deadline without an approved extension of time

will be penalised. The penalty for late submission is a reduction by 5% of the maximum

Assignment Mark, for each University Business Day or part day that the Assignment is late.

An Assignment submitted more than ten University Business Days after the deadline will have

a Mark of zero recorded for that Assignment.

You must have completed the Academic Integrity training course before you submit this assign-

ment.

We expect a high standard of communication. Look at the worked examples in your texts, and

the sample solutions in Tutorial Worked Examples to see the level you should aim at. Up to

15% of the marks may be deducted for poor language and notation.

Please note: No Assignments for Assessment purposes will be accepted after as-

signment solutions have been released.

ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 2 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 2 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 2 of 4

QUESTION 1 (16 marks)

Find each of the following limits:

(a) lim

t→−7

5t2 + 24t− 77

4t2 + 35t+ 49

; (8 marks)

(b) lim

t→∞

420t3 − 7t4 − 28t9 + 18

(t6)

√

23

t

− 13t3 + 676t6 + t5

; (8 marks)

QUESTION 2 (14 marks)

A rocket is travelling in a straight line for a minute. The distance in metres covered by the rocket

during this time is described by the function

s(t) = 2t2 + 10t+ 128− 40 ln

(

t2 + 4

)

where t ≥ 0 and time is given in seconds.

(a) Find a function that describes the speed of the rocket. (3 marks)

(b) What is the speed of the rocket at the time t = 6 seconds? (1 mark)

(c) Find all values of time t (if any) when the speed of the rocket is 10 m s−1. (3 marks)

(d) Find a function that describes the acceleration of the rocket. (3 marks)

(e) Find the acceleration of the rocket at t = 4 seconds. (1 mark)

(f) Find all values of time t (if any) when the rocket’s acceleration is 4 m s−2. (3 marks)

QUESTION 3 (14 marks)

The position of a drone at time t (in minutes) is given by the parametric function

x = 5 tan−1 2t+ 7 e−7(t−1) sin

(

t2 − 3t+ 2

)

y = 6 ln (t XXXXXXXXXXe−7(t−1) cos

(

t2 − 3t+ 2

)

where both x and y are measured in metres.

(a) Find an expression for

dy

dx

in terms of t. (11 marks)

(b) Using part (a) evaluate the derivative when t = 1. (3 marks)

ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 3 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 3 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 3 of 4

QUESTION 4 (20 marks)

Mr Clarkson finds that the time taken to mow his lawn increases the longer he delays mowing. From

past experience he finds the time taken (in hours) to mow the lawn once, if he has not mown for t

weeks, is approximated by the function

C(t) =

4t

(

t2 − 5t+ 7

)

3t2 − 16t+ 24

.

(a) Find an expression for the total time spent mowing per year assuming he mows regularly every

t weeks.

You may assume there are 52 weeks in a year and the time taken to do the actual mowing is

incorporated into the value of t. (4 marks)

(b) If Mr Clarkson is to mow the lawn on a regular basis (every t weeks) determine, using calculus,

the optimal length of time should he delay mowing to minimise the total time per year he has

to mow.

Check your value of t by substitution into the derivative and verify that you have found the

minimum by using an appropriate calculus test.

What is the minimum total time spent mowing per year? (16 marks)

QUESTION 5 (16 marks)

The speed of a car at time t (in seconds) is given by a piecewise function V (t) (in m/s) as shown

below. Determine the total distance, d, travelled by the car from t = 0 seconds to t = 34 seconds.

t (s)

V (m/s)

V (t) =

7 t2

√

t

81

, if 0 ≤ t ≤ 9,

21− 11

π

cos

(

πt

2

)

, if 9 ≤ t ≤ 19,

21 exp

(

−(t− 19)

3

)

, if 19 ≤ t ≤ 34.

XXXXXXXXXX 32

4

8

12

16

20

24

Note: the exponential function, ex, can be written as exp (x).

ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 4 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 4 of 4ENM1600 — Engineering Mathematics Assignment 3 Due: Tuesday 25 May 2021 Page 4 of 4

QUESTION 6 (20 marks)

To help find the velocity of particles requires the evaluation of the indefinite integral of the acceleration

function, a(t), i.e.

v =

∫

a(t) dt.

Evaluate the following indefinite integrals.

Check your value for each integral by differentiating your answer.

(a)

∫ (

27t2 − 12t− 9

√

t

)

ln t dt; (8 marks)

(b)

∫

7t2

(

27 cos 3t− 8e−2t

)

dt; (12 marks)

End of Assignment XXXXXXXXXXmarks Total)

Answered 6 days AfterMay 17, 2021

Cam...

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