Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#7

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Question

The diagram shows a metal plate ABCDEF which has been made by removing the two shaded regions from a circle of radius 10 cm and centre O. The parallel edges AB and ED are both of length 12 cm.

    i.        Show that angle DOE is 1.287 radians, correct to 4 significant figures.

   ii.       Find the perimeter of the metal plate.

  iii.       Find the area of the metal plate.

Solution

    i.
 

Consider the diagram below.

Line OG bisects both the angle DOE and the chord DE. If we consider the triangle OGD, we can write;

Expression for   trigonometric ratio in right-triangle is;


ii.
 

It is evident from the diagram that; 

We are given that;

Now we find out length of arcs BCD & EFA.

Expression for length of a circular arc with radius  and angle  rad is;

For the given case;

It is evident from the diagram below that;

Hence;

Similarly, by symmetry, we can find;

Therefore;


iii.
 

It is evident from the given diagram that; 

Expression for area of a circle with radius  and angle  rad is;

For the given case;

It is evident from the diagram that; 

Expression for area of a circular sector with radius  and angle  rad is;

Therefore;

Expression for the area of the triangle is;

Consider the .

Pythagorean Theorem

Therefore;

Hence,

By symmetry, area of other shaded region is also same i.e.16.35.

Finally; 

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