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:
5. To find a formula for the area of the circle, find the limit of the expression in step 4 as
Trending nowThis is a popular solution!
Chapter 2 Solutions
Calculus Volume 1
Additional Math Textbook Solutions
Calculus Volume 2
Introductory Statistics
Using & Understanding Mathematics: A Quantitative Reasoning Approach (7th Edition)
Probability and Statistics for Engineers and Scientists
Thinking Mathematically (7th Edition)
Mathematics for Elementary Teachers with Activities (5th Edition)
- Question 16arrow_forwardThe figure shows two circles C and D of radius 1 that touch at P. The line T is a common tangent line; C, is the circle that touches C, D, and T; C, is the circle that touches C, D, and C;; C3 is the circle that touches C, D, and C3. This procedure can be continued indefinitely and produces an infinite sequence of circles {C,}. Find an expression for the diameter of C, and thus provide another geometric demonstration of Example 8. In[n+1] dn = 1 C Tarrow_forwardQuestion 4. As an engineer, you need to design the cable supporting the bridge deck. An initial analysis shows that the concrete tower is 175 meters above the road deck and that the support point for the main cable 125 meters away from the concrete tower. How long does the cable have to be?arrow_forward
- A rectangle is to be inscribed in a semicircle of radius B, as shown in the figure. What is the largest possible area of such a rectangle? (Hint: The width of the rectangle is the second coordinate of the point P (why?), and Pis on the top half of the circle x² + y264.1arrow_forwardMany smartphones are now able to use information sent out by GPS satellites (of which there are about nine overhead at any one time) in a similar way. Since the satellites are moving around in space, and therefore can’t be located on a two-dimensional map of the Earth, we now need to consider three-dimensional geometry. How many satellites are needed to be sure of the location of a smartphone? Explain your findingsarrow_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
- 2) The point P(x, –) lies on the unit circle in quadrant III. Find its x co-ordinate. Page 1 of 5arrow_forwardQuestion 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_forwardTwo railroad tracks intersect at right angles. At noon there is a train on each track approaching the crossing at 40 mi. per hr., one being 100 mi., the other 200 mi. distant. Finda. When they will be nearest together; and;b. What will be their minimum distance apart?arrow_forward
- The two-point Gaussian quadrature formula gives an exact value for , (2x – 7x3 + 5) dx, True Falsearrow_forwardQuestion 9 Compute 14 x 5 x (15 – 9) – 9 +8 350 O 419 O 1040 1024arrow_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
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,