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October 22nd, 2013, 11:04 AM   #1
kot
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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 x-y=-30 and x+3-y(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
x-y=-30 worked for both solutions but x+3-y(squared)=189 did not work with (12, 42)

2.)A 250-g 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)(d-20)(squared) and its potential energy, E(p), is given by the equation E(p)=(-5/32)(d-20)(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
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October 22nd, 2013, 11:23 AM   #2
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Re: Systems and Solving Inequalities in Two Variables

Applications of Systems
1.
a) a - b = -30, b + 3 + a = 189.
b) 30 + a + 3 + a = 189, a + a - 156 = 0, a = 12, -13.
c) b = 30 + a, b + 3 + a = 189 for both values of a.
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October 23rd, 2013, 06:09 AM   #3
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Re: Systems and Solving Inequalities in Two Variables

Quote:
Originally Posted by kot
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 x-y=-30 and x+3-y(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
x-y=-30 worked for both solutions but x+3-y(squared)=189 did not work with (12, 42)
They don't satisfy your equations because you have the wrong equations. You did not think carefully about what your "x" and "y" represent. You write "x+ 3- y^2= 189" for "When the larger integer is increased by 3 and added to the square of the smaller integer, the result is 189" so you must mean x to be the larger integer, y the smaller. But in that case "x- y= -30" is impossible. A larger number minus a smaller is positive, not negative. Either you must have y- x= -30 or you copied the problem incorrectly and the difference is "30" not "-30".

Quote:
2.)A 250-g 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)(d-20)(squared) and its potential energy, E(p), is given by the equation E(p)=(-5/32)(d-20)(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)
Set the two formulas for energy equal, (5/32)(d- 20)^2= (-5/32)(d- 20)^2+ 62.5, and solve for d.

Quote:
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)
No, if the cost is 300x+ (1/12)x^2 then the marginal cost must be 300+ (1/6)x.

Quote:
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)
Great! I'd like to see how you got that since setting 50000- 20x equal to either "300+ (1/4)x^2" or "300+ (1/6)x", I do NOT get an integer solution.

Quote:
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
5000x- 10x^2 is the revenue but 300+ (1/4)x^2 is NOT the cost. Try 5000x- 10x^2- (300x+ (1/12)x^2)= 4700x - (121/12)x^2. You can find the maximum value of that by completing the square.

Quote:
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)[quote:3sgeqaul]
The vertices will be where those parabolas cross so at (x, y) values thaat solve two of them. Solve
y= x^2+ 8x+ 16= x^2- 8x+ 16
y= x^2+ 8x+ 16= -x^2/8+ 2 and
y= x^2- 8x+ 16= -x^2/8+ 2

[quote:3sgeqaul]b)Estimate the area of material required to make the sail.
(Example: 50m(squared))
Since this says "estimate" I would graph it and then fit rectangles in it.

Quote:
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
[/quote:3sgeqaul][/quote:3sgeqaul]
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.
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October 23rd, 2013, 04:51 PM   #4
kot
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Re: Systems and Solving Inequalities in Two Variables

Quote:
They don't satisfy your equations because you have the wrong equations. You did not think carefully about what your "x" and "y" represent. You write "x+ 3- y^2= 189" for "When the larger integer is increased by 3 and added to the square of the smaller integer, the result is 189" so you must mean x to be the larger integer, y the smaller. But in that case "x- y= -30" is impossible. A larger number minus a smaller is positive, not negative. Either you must have y- x= -30 or you copied the problem incorrectly and the difference is "30" not "-30".
I found my mistake was quickly reading through the question. I had created the equation "x + 3 - y^2 = 189" when the equation should have been "x + 3 + y^2 = 189" and the mistake had been carried on throughout my work which made it unable to solve.
Quote:
Set the two formulas for energy equal, (5/32)(d- 20)^2= (-5/32)(d- 20)^2+ 62.5, and solve for d.
I worked under the assumption that E(k) and E(p) were two different variables therefore I had not thought of placing them in an equation by making both systems equal. After making the two equations equal I was able to isolate d and answer the question with the correct answer.
Quote:
No, if the cost is 300x+ (1/12)x^2 then the marginal cost must be 300+ (1/6)x.
I rechecked the equations to see if perhaps I miss typed but the given equations were not wrong.
Quote:
Great! I'd like to see how you got that since setting 50000- 20x equal to either "300+ (1/4)x^2" or "300+ (1/6)x", I do NOT get an integer solution.
I took R(m), the marginal revenue, and made it equal to C(m), the marginal cost.
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:
5000x- 10x^2 is the revenue but 300+ (1/4)x^2 is NOT the cost. Try 5000x- 10x^2- (300x+ (1/12)x^2)= 4700x - (121/12)x^2. You can find the maximum value of that by completing the square.
I'm a bit unclear how you got the equation "5000x- 10x^2- (300x+ (1/12)x^2)= 4700x - (121/12)x^2". Neither the less, I tried to shift over the equation and collect like terms
5000x-10x^2-300x-(1/12)x^2-4700+(121/12)x^2=o
And I came up with the binomial
4700x-4700=0
Leading me to become unclear how to complete the square
Quote:
The vertices will be where those parabolas cross so at (x, y) values thaat solve two of them. Solve
y= x^2+ 8x+ 16= x^2- 8x+ 16
y= x^2+ 8x+ 16= -x^2/8+ 2 and
y= x^2- 8x+ 16= -x^2/8+ 2
Thank-you so much for the equations, once those were written down the question made sense to me
Quote:
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.
I remembered dealing with inequalities in previous years but I guess the crucial piece of information was forgotten. Thanks for explaining why the sign switched.
Quote:
In future- do not post so many problems at once and show what you have tried yourself.
Will do! ^^
Sorry for the inconvenience of throwing multiple questions without work shown into a post.
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