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October 22nd, 2013, 11:04 AM  #1 
Newbie Joined: Feb 2012 Posts: 4 Thanks: 0  Systems and Solving Inequalities in Two Variables
Due to being away from school I have had to teach myself the lessons I missed without the notes necessary and I was hoping that someone could help me out by explaining how they got the answer. Applications of Systems 1.)Two integers have a difference of 30. When the larger integer is increased by 3 and added to the square of the smaller integer, the result is 189/ a) Model the given information with a system of equations I got xy=30 and x+3y(squared)=189 b)Determine the value of the integers by solving the system. The answers sheet has (12,42) and (13,17)~did not get those answers c)Verify your solution I substituted the x and y values that the book provided, however, in both equations I did not get the left side to equal the right side xy=30 worked for both solutions but x+3y(squared)=189 did not work with (12, 42) 2.)A 250g ball is throws into the air with an initial velocity of 22.36m/s. The kinetic energy, E(k), of the ball is given by the equation, E(k)=(5/32)(d20)(squared) and its potential energy, E(p), is given by the equation E(p)=(5/32)(d20)(squared)+62.5, where energy is measured in joules (J)and d is the horizontal distance traveled by the ball, in metres. a)At what distance does the ball have the same amount of kinetic energy as potential energy? (answer should be 5.86 m and 34.14 m) b.) How many joules of each type of energy does the ball have at these distances? (answer should be 31.25 J) 3.)The monthly economic situation of a manufacturing firm is given by the following equations. R= 5000x  10x(squared) R(m) = 5000  20x C= 300x + (1/12)x(squared) C(m)= 300 + (1/4)x(squared) Where x represents the quantity sold, R represents the firm's total revenue, R(m) represents the marginal revenue, C represents total cost, and c(m) represents the marginal cost. All costs are in dollars. a)Maximum profit occurs when marginal revenue is equal to marginal sold. How many items should be sold to maximize profit? The correct number of items is 103 (which I got) b)Profit is total revenue minus total cost. What is the firm's maximum monthly profit? I tried to make the equation p= 5000x  10x(squared)  300  (1/4)x(squared) but I did not manage to get the correct number which is $377,125.92 4.)Kate is an industrial design engineer. She is creating the program for cutting fabric for a shade sail. The shape of a shade sail is defined by three interesting parabolas. The equation of the parabolas are y= x(squared) + 8x + 16 y= x(squared)  8x + 16 y= (x(squared)/ + 2 Where x and y are measured in metres. a)Use an algebraic method to determine the coordinates of the three vertices of the sail (answer should be (3.11,0.79), (3.11,0.79) and (0,16). I've never worked with three equations at once so I did not know how to even begin the question) b)Estimate the area of material required to make the sail. (Example: 50m(squared)) Solving Inequalities in Two Variables 1.)Determine the solution to 5y(is less than or equal to)x I managed to isolate the y value 5y(is less than or equal to)x y(is less than or equal to) (1/5)x However the answer is y(is greater than or equal to) (1/5)x and so I became confused how the sign switched 
October 22nd, 2013, 11:23 AM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond  Re: Systems and Solving Inequalities in Two Variables Applications of Systems 1. a) a  b = 30, b + 3 + a² = 189. b) 30 + a + 3 + a² = 189, a² + a  156 = 0, a = 12, 13. c) b = 30 + a, b + 3 + a² = 189 for both values of a. 
October 23rd, 2013, 06:09 AM  #3  
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Systems and Solving Inequalities in Two Variables Quote:
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Notice that 3< 5 but (multiplying by 3), 9> 15. When you multiply or divide an inequality by a negative number, the order is reversed. In future do not post so many problems at once and show what you have tried yourself.  
October 23rd, 2013, 04:51 PM  #4  
Newbie Joined: Feb 2012 Posts: 4 Thanks: 0  Re: Systems and Solving Inequalities in Two Variables Quote:
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5000  20x = 300 + (1/4)x^2 0 = (1/4)x^2 + 20x  4700 I substituted the trinomial into a quadratic equation http://upload.wikimedia.org/math/3/c...3ccad7990f.png A= (1/4)x^2 B= 20x C=  4700 After substituting the values I got 102.8285686. In this case I assumed it was rounded up to 103 despite the fact that in a real life case(and other word problems) it would be most probable to round down. Quote:
5000x10x^2300x(1/12)x^24700+(121/12)x^2=o And I came up with the binomial 4700x4700=0 Leading me to become unclear how to complete the square Quote:
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Sorry for the inconvenience of throwing multiple questions without work shown into a post.  

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