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# Equitlateral Triangles within a Closed Area

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(a) Figure 1 shows a closed area ABCDEF in which ABDE is a rectangle and BCD and AFE are equilateral triangles. AE x cm and AB y cm.
(i) Find, in terms of x andy, a formula for the area enclosed by the figure ABCDEF and a formula for the perimeter ABCDEF.
(ii) Find the minimum perimeter (to two decimal places) of ABCDEF enclosing an area of 400 cm2.
(iii) Find the maximum area (to two decimal places) ofABCDEF enclosed by a perimeter of 400 cm,
Figure 1
(b) Prove that, when x 1, there are three points on the curve xy3 + 2x = 2x2y2 + y. Find the corresponding y-coordinates of these points and the equations of the tangent and normal at each of these points.
(c) Find the equations of the tangent and normal at a general point (with parameter t) for the cycloid x = t ? cost ?2; y = 1 + 2 sin t . Use Derive to 2D-plot the curve for ?3 < t < 8 and, using Derive VECTOR commands, set up and 2D-plot the tangent and normal families of the cycloid for ?3m< t < 8 (in steps of 0.2). Submit Derive printouts (including relevant algebraic and graphical results).

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Equitlateral Triangles within a Closed Area are investigated. Th solutio is detailed and has a diagram.

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(1) Denote the area and the perimeter of ABCDEF by S and P. Then we have

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###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
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• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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