this way] The cost C(x) of the pipeline as a function of x is: C(x) = distance along north shore * pipeline cost over land + distance under the river * pipeline cost under land The distance along the north shore is 6-x The distance (by Pythagorean theorem) under the water is sqrt( 2^2 + x^2) So‚ C(x) = (6-x)*200000 + sqrt(4 + x^2) * 400000 [You should graph this] To find the value of x where C(x) is minimized‚ we set dC/dx = 0‚ [Reminder - use the chain rule to differentiate the
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MATH133 UNIT 2: Quadratic Equations Individual Project Assignment: Version 2A Show all of your work details for these calculations. Please review this Web site to see how to type mathematics using the keyboard symbols. Problem 1: Modeling Profit for a Business IMPORTANT: See Question 3 below for special IP instructions. This is mandatory. Remember that the standard form for the quadratic function equation is y = f (x) = ax2 + bx + c and the vertex form is y = f (x) = a(x – h)2 + k‚ where (h‚ k) are
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10. ’Through different methods of justification‚ we can reach conclusions in ethics that are as well-supported as those provided in mathematics.’ To what extent would you agree? One could argue that mathematics and ethics are the underlying essentials above which our society has based itself. Scores of cities have built their infrastructures using measurements and methods founded in mathematics. Our inherent ethical natures have catalyzed the great minds from ancient civilizations to create democracies
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03.01LessonSummary To achieve mastery of this lesson‚ make sure that you develop responses to the essential question listed below. How can a Greatest Common Factor be separated from an expression? By simplifying the equation . By breaking them up by dividing them up What methods can be used to rewrite square trinomials and difference of squares binomials as separate factors? distribution in what conditions can a factored expression be factored further? Greatest Common Factor A greatest common factor
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With the help of tape measures and rules small distances in the everyday household are easy to measure. As distances grow bigger so do the devices and ways in which the distance is to be measured. In math rulers make sense as well as Pythagorean Theorem; so what about astronomy? Astronomers have a totally different format of information of which they are studying. It only makes sense that the astronomers would have a totally different way of which they measure. Two measurements astronomers use are
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Completing the Square: Quadratic Examples & Deriving the Quadratic Formula (page 2 of 2) Solve x2 + 6x + 10 = 0. Apply the same procedure as on the previous page: This is the original equation. x2 + 6x + 10 = 0 Move the loose number over to the other side. x2 + 6x = – 10 Take half of the coefficient on the x-term (that is‚ divide it by two‚ and keeping the sign)‚ and square it. Add this squares value to both sides of the equation. x^2 + 6x + 9 = –10 + 9 Convert the left-hand side to
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side to the other two sides and its corresponding angle: Relate that either of these relations reduce to simpler forms for the case of right triangles‚ particularly: 1. Law of Sines reduces to c = c = c. 2. Law of Cosines reduces them to the Pythagorean Theorem. Solving Oblique Triangles Explore the types of triangles and how to solve them. All oblique triangles make full use of the sum of angles rule that A + B + C = 180º in all triangles. All other elements require select application of
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------------------------------------------------- Spiral of Theodorus In geometry‚ the spiral of Theodorus (also called square root spiral‚ Einstein spiral or Pythagorean spiral)[1] is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene. ------------------------------------------------- ------------------------------------------------- [edit]Construction The spiral is started with an isosceles right triangle‚ with each leg having a length of 1
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have been the first person to produce a table for solving a triangle’s lengths and angles.[2] ------------------------------------------------- [edit]The Pythagorean Theorem Pythagoras‚ depicted on a 3rd-century coin In a right triangle: The square of the hypotenuse is equal to the sum of the squares of the other two sides. The Pythagorean theorem is named after the Greek mathematician Pythagoras‚ who by tradition is credited with its discovery and proof‚[1][2] although it is often argued that
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includes an equation to determine whether a triangle is or isn’t right-angled. Haylock (2010) also identifies a concept‚ as the Pythagorean triple as three natural numbers that could be the lengths of the three sides of a right-angled triangle. For example‚ 5‚ 12 and 13 form a Pythagorean triple because 52 + 122 = 132. Other well known examples of Pythagorean tripes are 3‚ 4‚ 5 (because 9 + 16 = 25) and 5‚ 12‚ 13 (because 25 + 144 = 169). Discussion: As mentioned
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