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Consider the function g defined by g(x)=2−4sin2x. The following graph shows the curve y=g(x) for 0<x<23π.
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The graph of y=g(x) intercepts the x-axis at a, b, c and d, as shown.
Find the values of a,b,c and d.
In the diagram above, the region enclosed by the curve of y=g(x) and the x-axis between x=b and x=c is shaded.
Show that the area of the shaded region is 34π+2√3.
The diagram below shows a circle of radius r circumscribing a shape whose area is 34π+2√3.
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Find the value of r.
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A right cone with slant length 5cm and diameter 8cm is placed with its vertex pointed straight down, and filled with water to a depth of h.
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Show that the volume of water in the cone is V=2716πh3.
Water is being added to the cone at a rate of 3cm3/s.
Find the rate at which h is increasing when V=15cm3.
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The function f is defined by f(x)=sin(ax+k), where a,k>0. The graph of y=f(x) is plotted below for −2b<x<4b, where x is in radians.
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Give an expression for b in terms of a.
Hence find the smallest possible value of k.
The function g is defined by g(x)=sin(32ax+k) for −3b<x<6b.
Sketch the graph of y=g(x) on the axes above.
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