MAS182: Applied Mathematics
Semester 1, 2013
Assignment 3
Due by: 4:00pm Friday, 12 April 2013
1. [8 marks] Sand is leaking from the back of a dump truck and forming a conical pile on the ground. The sand is leaking at the rate of 0.2 m3 per hour. If the base radius of the pile is always 0.6 times the height, how fast is the base radius changing when the height
is 1.5 m? Show all the working.
(Volume of cone of heighth and base radiusr isV =13 r2h.)
p
2. [7 marks] Let f (x) = 5 + (x2 x 3) ln( x 1)
(a) Find the derivative off .
(b) Using the derivative and linear approximation, estimatef (4:08).
3. [16 marks] Find the following integrals, showing all working.
?
(a) x(3×2 1)1=2 dx
?
(b) 3x(4x + 2)?2dx
? 8
(c) (t2=3 + 4e?t=2)dt
?01
(d) 4x2e2?x3dx
0
4. [6 marks] A dam is overflowing at the rate
V?(t) =
3t
thousand m3
per day;
(t2
+ 1)2=3
wheret is the time in days after the overflow begins. Find the amount of water that overflows from the dam during the second day of the overflow. Give the answer exactly first and then to 4 significant figures.
p
5. [8 marks] Sketch the region bounded byf (x) =x + 1,g(x) =e?2x,x = 1 andx = 2. Using calculus, find the area of the region, showing all the working. Express your answer in simplified exact form.
Notes:
10% of the marks for this assignment are reserved for presentation.
There are penalties for late assignments. You must contact your tutorbefore the due date if you have di?culties making the deadline.
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