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ENM1600 Engineering Mathematics, S1–2020 Assignment 3
Value: 10%. Due Date: Tuesday 26 May 2020.
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QUESTION 1 (16 marks)

Find each of the following limits:
(a) ; (8 marks)
(b) . (8 marks)
QUESTION 2 (14 marks)

A rocket 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

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 = 10 seconds? (1 mark)
(c) Find all values of time t (if any) when the speed of the rocket is 50ms-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 = 18 seconds. (1 mark)
(f) Find all values of time t (if any) when the rocket’s acceleration is zero i.e. 0ms-2. (3 marks)
QUESTION 3 (14 marks)

On a fun park ride the position of the carriage at time t (in minutes) is given by the parametric
function

where both x and y measured from the axis of rotation and are given in metres.
(a) Find an expression for in terms of t. (11 marks)
(b) Using part (a) evaluate the derivative when t = 0. (3 marks) QUESTION 4 (20 marks)

Two ropes are attached to the ceiling at points 20 cm apart. The rope on the left is 7 cm long and has a pulley at its end. The rope on the right, of length is 42 cm (shown in blue), passes through the pulley and has a weight attached to its end as shown below (not drawn to scale). At rest the ropes and pulley arrange themselves such that the vertical distance from the ceiling to the weight is maximized.

(a) Express the vertical distance from the ceiling to the weight as a function of x. (4 marks)
(b) Using Calculus find the value of x (in metres) that gives the maximum vertical distance from the ceiling to the weight.
What is the maximum vertical distance?
Check your value of x by substitution into the derivative. (10 marks)
(c) Confirm that you have found the maximum amount using an appropriate test. (6 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 = 24 seconds.

Note: the exponential function, ex, can be written as exp(x).
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. Z v = a(t)dt.
Evaluate the following indefinite integrals.