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Geometry
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Question:

For a triangle with vertices A (1,6), B (5,1), and C (1,1): a) find each side length and the perimeter b) find the area c) find the each of the three inside angles

English
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Question:

For the following passage a) denote the errors by putting them in brackets [] b) correct the passage. The well-known book the jungle book came into it's own as one of the animated highlights of the Disney empire. It was published in 1894, and its sequel The Second Jungle Book came a year later during the time when Kipling was ensconced at Brattleboro in Vermont. The storie tells of the child Mowgli who is a foundling brought up by wolfs. He learns over time and due to the instructions of varius animal mentors the rules or ‘Laws’ of the jungle. Key figures are the wise black panther, Bagheera, and Baloo the sleepy bear. Both of these friendly beasts contribute to the childs education. We learn about the great enmity between Mowgli and the tiger Shere Khan who killed the boy’s parent's. Like Just So Stories (1902) it portrays the natural world and especially its creatures in a logical anthropomorphized manner, entertaining to adult and childs alike. The simplicity of the concept and the lack of didactic moral overtones have made The Jungle Book a lasting influence on the young. (source: http://www.bibliomania.com/0/-/frameset.html)

a) The well-known book [the jungle book] came into [it's] own as one of the animated highlights of the Disney empire. It was published in 1894, and its sequel The Second Jungle Book came a year later during the time when Kipling was ensconced at Brattleboro in Vermont. The [storie] tells of the child Mowgli who is a foundling brought up by [wolfs]. He learns over time and due to the instructions of [varius] animal mentors the rules or ‘Laws’ of the jungle. Key figures are the wise black panther, Bagheera, and Baloo the sleepy bear. Both of these friendly beasts contribute to the [childs] education. We learn about the great enmity between Mowgli and the tiger Shere Khan who killed the boy’s [parent's]. Like Just So Stories (1902) it portrays the natural world and especially its creatures in a logical anthropomorphized manner, entertaining to adult and [childs] alike. The simplicity of the concept and the lack of didactic moral overtones have made The Jungle Book a lasting influence on the young. b) The well-known book The Jungle Book] came into its own as one of the animated highlights of the Disney empire. It was published in 1894, and its sequel The Second Jungle Book came a year later during the time when Kipling was ensconced at Brattleboro in Vermont. The story tells of the child Mowgli who is a foundling brought up by [wolfs]. He learns over time and due to the instructions of various animal mentors the rules or ‘Laws’ of the jungle. Key figures are the wise black panther, Bagheera, and Baloo the sleepy bear. Both of these friendly beasts contribute to the child's education. We learn about the great enmity between Mowgli and the tiger Shere Khan who killed the boy’s parents. Like Just So Stories (1902) it portrays the natural world and especially its creatures in a logical anthropomorphized manner, entertaining to adult and child alike. The simplicity of the concept and the lack of didactic moral overtones have made The Jungle Book a lasting influence on the young.

Algebra
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Question:

For the following equation: a) put it in standard form b) find the vertex c) find the y-intercept and critical points. y=2x^2+12x+17

a) This is a quadratic function, which we know because it has the form y=ax^2+bx+c. The standard form of a quadratic function is y=a(x-h)^2+k. We can change the given function to its standard form by completing the square. Follow these steps: y=2x^2+12x+17 The coefficient on x^2 must be 1, so we need to factor the 2 out of both x-terms. y=2(x^2+6x)+17 Now we will complete the square. To do this, we have to find a number that we can add to (x^2+6x) that will make it equal to some square in the from of (x+a)^2. We must remember, though, that a basic rule of algebra is "whatever you do to one side of the equation, you must do to the other." So, when we add this number to the right side of the equation, we must add it to the left side as well. It is important to note that this new number will be inside a set of parentheses that is multiplied by 2. This means that the number will also have to be multiplied by 2 when we add it to the other side. This is the "complete the square" equation (the blanks represent where we will add the new number): y+2(__)=2(x^2+6x+__)+17 Finding the number we will add is simple. We use the term 6x. The coefficient of this term is 6. Divide it by 2, and square the answer. This will be our new number 6/2=3 3^2=9 Putting 9 in the blanks, we now have: y+2(9)=2(x^2+6x+9)+17 Now we simplify. First, factor x^2+6x+9. We know that we square 3 to get 9 and complete the square (above). Therefore, we can know that x^2+6x+9=(x+3)^2. Second, we will do 2(9) and move the result to the right side of the equation. 2(9)=18, so subtract 18 from both sides to get y by itself. x^2+6x+9=(x+3)^2 2(9)=18 y+18=2(x+3)^2+17 y+18-18=2(x+3)^2+17-18 y=2(x+3)^2-1 We now have the standard form: y=2(x+3)^2-1 b) We can find the vertex and y-intercept very easily from the standard form of the equation. We know the general standard form of quadratic functions: y=a(x-h)^2+k We have: y=2(x+3)^2-1 From those, we can see that: a = 2 h = -3 k = -1 For quadratic equations in the standard form, it is known that the vertex is at the point (h,k). Therefore, our vertex is (-3,-1). c) To find the y-intercept and critical points, we will use the original form of the equation, y=2x^2+12x+17. For the y-intercept, set x=0 and solve for y. Remember that anything multiplied by 0 equals 0. y=2(0)^2+12(0)+17 y=0+0+17 y=17 Therefore, our y-intercept is at the point (0,17). This is because we used x=0 and found y=17. For the critical points, we will have to find the roots of the equation. Because this equation cannot be factored easily, we will use the quadratic equation: 1: x = [ -b + sqrt( b^2 - 4ac )] / [2a] 2: x = [ -b - sqrt( b^2 - 4ac )] / [2a] Because our original equation is in the form of y=ax^2+bx+c, we can know: a = 2 b = 12 c = 17 Now we plug these numbers into our two quadratic equations to find the two critical points. 1: x = [ -12 + sqrt( 12^2 - 4(2)(17) )] / [2(2)] x = { -12 + sqrt( 144 - 136)] /  x = [ -12 + sqrt( 8 )] /  sqrt(8) = sqrt(4) * sqrt(2) = 2 * sqrt(2) -12/4= -3 2*sqrt(2)/4 = .5 * sqrt(2) x = -3 + .5 * sqrt(2) 2: x = [ -12 - sqrt( 12^2 - 4(2)(17) )] / [2(2)] x = { -12 - sqrt( 144 - 136)] /  x = [ -12 - sqrt( 8 )] /  sqrt(8) = sqrt(4) * sqrt(2) = 2 * sqrt(2) -12/4= -3 2*sqrt(2)/4 = .5 * sqrt(2) x = -3 - .5 * sqrt(2) Because critical points are the points at which the line crosses the x axis, they will be at y=0. Therefore, our two critical points are: ( [-3 + .5 * sqrt(2)] , 0 ) ( [-3 - .5 * sqrt(2)] , 0 ) Summarizing our answers, we have: a) y=2(x+3)^2-1 b) vertex is (-3,-1). c) y-intercept (0,17) critical points are ( [-3 + .5 * sqrt(2)] , 0 ) and ( [-3 - .5 * sqrt(2)] , 0 )

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