MAS182: Applied MathematicsSemester 1, 2013Assignment 3Due by: 4:00pm Friday, 12 April 20131.[8 marks] Sand is leaking from the back of a dump truck and forming a conical pile onthe ground. The sand...

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 3 the ground. The sand is leaking at the rate of 0.2 m 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. 1 2 (Volume of cone of height h and base radius r is V = r h.) 3 p 2 2. [7 marks] Let f(x) = 5+(x x3)ln( x1) (a) Find the derivative of f. (b) Using the derivative and linear approximation, estimate f(4:08). 3. [16 marks] Find the following integrals, showing all working. ? 2 1=2 (a) x(3x 1) dx ? -2 (b) 3x(4x+2) dx ? 8 2=3 -t=2 (c) (t +4e )dt 0 ? 1 3 2 2-x (d) 4x e dx 0 4. [6 marks] A dam is over?owing at the rate 3t ' 3 V (t) = thousand m per day; 2 2=3 (t +1) where t is the time in days after the over?ow begins. Find the amount of water that over?ows from the dam during the second day of the over?ow. Give the answer exactly ?rst and then to 4 signi?cant ?gures. p -2x 5. [8 marks] Sketch the region bounded by f(x) = x+1, g(x) = e , x =1 and x = 2. Using calculus, ?nd the area of the region, showing all the working. Express your answer in simpli?ed exact form. Notes:  10% of the marks for this assignment are reserved for presentation.  There are penalties for late assignments. You must contact your tutor before the due date if you have di?culties making the deadline. 1