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...

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

Solution

Sultana answered on May 23 2021
20 Votes

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