Problem Statement
1.Trajectory of a rocket
Consider a rocket initially located at the origin: (0,0,0). This rocket is fitted with a simple transmitting device that relays information about the rocket’s movement. Each transmission contains a vector describing the movement of the rocket, relative to the position of the rocket at the previous transmission. The rocket returns the following transmissions in order (a then b then c). All transmissions are in kilometres. We are not interested in what happens to the rocket after the third transmission:
(a) Suppose that at the time of the third transmission, the rocket is located at some point P. Determine the vector, −−→OP describing the position of the rocket relative to the origin, AND determine the distance of the rocket from the origin
(b) Assume that the earth is flat, and that the rocket follows a straight trajectory between transmissions. Calculate the pitch angle of the rocket (the angle the rocket makes with the xy-plane) at the time of the third transmission
(c) Calculate the deflection of the rocket’s heading on the ground between the first and the third transmissions (i.e., the change in direction seen on the ground)
2.Chemical Catastrophe
You work as a chemical engineer, your job is to combine various chemicals to produce pharmaceutical medicine. One day, your boss cannot make it to work, so you are in charge.
You get a panicked phone-call from a hospital; the hospital urgently needs their delivery of Medicine X. Unfortunately, you don’t have any of this particular medicine ready, and it’s one that you have not made before (normally your boss handles Medicine X ). Because of this, you are not sure how it is made (you are not sure how much of which chemicals mix together to produce Medicine X ).
You look through the documents your boss has that are relevant to this client. Although you don’t find a recipe for Medicine X, you do find the electronic accounting ledger where your boss has calculated the cost of producing Medicine X the last three times it was produced (most recent entry in row 1):
Hypocycloid
It is noted in the ledger that each chemical’s cost is given in dollars per unit of chemical. You also remember your boss ranting about how chemical prices kept changing, making it hard for him to keep
profit predictable. You also note that the hospital ordered the same quantity of Medicine X each time, which is the same quantity that they want now.
From the ledger, you can see that Medicine X is produced using chemical x, chemical y and chemical z.
(a) By forming an appropriate matrix system and applying Gaussian Elimination to an augmented matrix, show that the required quantities of chemical x, chemical y and chemical z are 2500, 1000 and
3000 respectively .
(b) Once the boss returns, they commend you on your great work. Your boss never had the time to make more than one batch of Medicine X at a time. Your facility allows you to produce batches 5 or 10
times larger than your current batch size. They note that if you made larger batches the company could negotiate better prices from their chemical suppliers. They estimate the following discounts on the most
recent ledger prices (first row of ledger):
• 25% off the price of each chemical (x, y and z) if purchasing five times the original amount of each,
• 30% off the price of each chemical (x, y and z) if purchasing ten times the original amount of each.
If you make larger batches, you will need to add a preservative to the product to increase the shelflife. The preservative costs $10,000 per unit. Based on how frequently Medicine X is ordered, the boss
determines that the preservative must be added as follows:
• 2 units of preservative if producing five times the original amount of Medicine X,
• 5 units of preservative if producing ten times the original amount of Medicine X
i) Complete the following table describing the costs associated with producing different amounts of
Medicine X
Batch size 1 5 10 |
Total cost of chemicals |
Total cost of preservatives |
Total cost of production |
ii) Based on this information, determine the most cost effective batch size
3. Revenge of the Cycloid
In tutorials, you studied the parametric equations for a cycloid (Vectors worksheet, problem solving exercise 3). These equations describe the path of a point on the radius of a disc which rolls without slip along a horizontal line. The parametric equations for the cycloid are:
r(t) = Σx(t)Σ = Σ t − sin(t) Σ
y(t) 1 − cos(t)
As an extension to this problem, we could consider the hypocycloid; a hypocycloid is the resulting curve traced by a point on the circumference of a circle that rolls without slip around the interior of another circle (see schematic). The position vector for a hypocycloid traced out by point P which sits on the circumference of a circle of radius r (with centre C), that rolls without slip on the circumference of the larger circle of radius R, is given by:
a) Explain (using words and mathematics) why the hypocycloid is a horizontal segment when R = 2r
b) Find vectors describing the velocity and acceleration of the point P on the hypocycloid for general values of r and R
c) When R = 2r, show that the acceleration of the hypocycloid is proportional to the distance of P from O .
1.a)
b)
c)
2.a)
We solve the equations using Gaussian elimination as follows:
Subtracting equation 1 from equation 2 we obtain
Then, subtracting equation 3 from equation 2 we get
Subtracting equation 5 from equation 4 we obtain
Substituting into equation 5 we get
Substituting and into equation 1 we get
b.i)
Batch size |
Total Cost of Chemicals |
Total Cost of Preservative |
Total cost of production |
1 |
19000 |
0 |
19000 |
5 |
= |
||
10 |
= |
b. ii)
3.a)
b)
c)
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