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Concept explainers
Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. The Greek mathematician Archimedes (ca. 287—212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the
We can estimate the area of a circle by computing the area of an inscribed regular polygon. Think of the regular polygon as being made up of n triangles. By taking the limit as the vertex angle of these mangles goes to zero, you can obtain the area of the circle. To see this, carry out the following steps:
1. Express the height h and the base b of the isosceles triangle in Figure 2.31 in terms of
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Chapter 2 Solutions
Calculus Volume 1
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Calculus Volume 2
Introductory Statistics
Finite Mathematics & Its Applications (12th Edition)
Mathematics All Around (6th Edition)
Mathematics for Elementary Teachers with Activities (5th Edition)
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- Question 16arrow_forwardpart d to #2: #2: Ship A is 100 km north of ship B at 10 AM. Ship A is sailing east at 20 km/h, while ship B is sailing north at 10 km/hr.arrow_forwardQuestion 2 The following table of values gives a company's annual profits in millions of dollars. Rescale the data so that the year 2001 corresponds to x = 0. b) 0.80 Find the R² value for the quadratic model. a) 0.98 c) 0.85 d) 0.89 Year e) 0.87 f) None of the above Profits (in millions of dollars) 2001 2002 2003 2004 23.8 25.2 26.3 2005 28.9 27.6 2006 28.4 Review Laterarrow_forward
- Q.1: The length of rectangular piece of paper exceeds its breadth by 5cm. If a strip of 0.5cm wide be cut all round the piece of paper, the area of the remaining part would be 500 square cms. Find its original dimensions.arrow_forwardAn isosceles triangle with a base of length b and height of h has a rectangle inscribed. If the base of the isosceles triangle is centred on the origin, and the upper right corner of the rectangle is at (x, y), find an expression for the area of the rectangle, A(z), in terms of x, h, and b only. You may find the diagram helpful in solving the problem. You can drag the green dot to see how it affects the area. (x, y) h A(x) =arrow_forward2) The point P(x, –) lies on the unit circle in quadrant III. Find its x co-ordinate. Page 1 of 5arrow_forward
- Question 5 Suppose that you want to create a rectangular animal enclosure using 80 linear feet of fencing. You also want to incorporate a pre-existing stone wall running 40 feet in a straight line. |Stone Wall (40 ft) (a) Let x be the length of fencing used on the side containing the wall. Determine the side lengths for the remaining sides of the rectangle in terms of this x, keeping to the constraint that 80 feet of fencing available. you have (b) Determine the objective function A(x) for the area of the enclosure given the value of x. Determine the value of x that maximizes this function's output. (c) Based on the scenario, determine the appropriate restriction to the domain of A(x) and maximize area with respect to that restriction.arrow_forwardPart 1: A tower is 120 meters tall. A cable is attached from the top of the tower to a point on the ground that is 50 meters from the base of the tower. How long is the cable? (Drawing a picture might help.) Be sure to explain your work and show details. Part 2: Suppose the cable described in part 1 is removed and a new cable is attached from the top of the tower to a point on the ground closer to the base of the tower. You are told the cable is 110 meters long. Is there something wrong with this? Explain what you are thinking about the proposed length of the cable for part 2. (This part of the problem does not necessarily require computation.)arrow_forwardQuestion 7 A rancher wants to fence in an area of 1,500,000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use? Submit Question Question 8 Find the point on the line 5x + 3y +3=0 which is closest to the point (-2,-2). Submit Question Jump to Answer Question 9 Jump to Answer Determine two positive values such that the sum of the first number and five times the second number is 425 and whose product is a maximum. Enter the solutions using a comma-separated list. Determine the maximum value.arrow_forward
- A spinner has four sections, each with different point values and areas:The “Move Ahead 2 Space” section is 1/4 of the area of the spinner.The “Move Ahead 3 Spaces” section is 1/3 of the area of the spinner.The “Lose A Turn” section is 1/4 of the area of the spinner.The “Go Back 2 Spaces” section is 1/6 of the area of the spinner.arrow_forwardAfter a dilation centered at the origin, the image of AB¯¯¯¯¯¯¯¯ is A′B′¯¯¯¯¯¯¯¯¯¯. If the coordinates of the endpoints of these segments are A(6, −4), B(2, −8), A′(9, −6), and B′(3, −12), what is the scale factor of the dilation?arrow_forwardSolve it corrects and explainsarrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,
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