# Answer on Abstract Algebra Question for lolo

Question #4366

I have to use mathcad to solve this question A ﬂexible wire P QRS, of total length 12 metres, is bent into a three-edged planar shape, and its ends P, S are placed against a disc of radius 9 metres with centre O, as shown in the diagram below. (The arc P S is not part of the wire.) The end-segments P Q and RS of the wire lie along straight lines through O, while the arc QR forms part of a circle with centre O and subtends an angle x (in radians) at O. This question concerns the area A enclosed between the wire and the edge of the disc, which is shown shaded below. This area can be expressed by A = f(x), where f(x)=9x(4 − 3x)(16 + 3x) / 2(2+x)^2 0 ≤ x ≤4/3 (a) (i) Plot the graph of the function f(x). Your graph should cover the interval [0, 1.33] in the x-direction and [0, 20] in the y-direction. (ii) By using the ‘Trace’ facility (and also, if you wish, the ‘Zoom’ facility), estimate to two decimal places the coordinates of the point on this graph at which y = f(x) takes its maximum value. (iii) On the same graph, plot the line y = 8. Using the ‘Trace’ facility, estimate to two decimal places both solutions of the equation f(x) = 8. (These solutions give the values of x for which the shaded area is 8 m2 .)

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