# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/12) | Q#5

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Question

The diagram shows a metal plate OABC, consisting of a right-angled triangle OAB and a sector  OBC of a circle with centre O. Angle AOB = 0.6 radians, OA = 6 cm and OA is perpendicular to OC.

i.       Show that the length of OB is 7.270 cm, correct to 3 decimal places.

ii.       Find the perimeter of the metal plate.

iii.       Find the area of the metal plate.

Solution

i.

We are required to show length of OB=7.270 cm.

It is evident from the diagram that OB is the hypotenuse of the right-angled triangle OAB.

Expression for  trigonometric ratio in right-triangle is;

For the given right-angled triangle OAB.;

ii.

It is evident from the diagram that for given metal plate;

Let’s find all these elements one-by-one.

We are given the first on; .

Next we have to find AB.

It is evident from the diagram that AB is the perpendicular of the right-angled triangle OAB.

Expression for  trigonometric ratio in right-triangle is;

For the given right-angled triangle OAB.;

Next we need to find length of arc BC.

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

Arc BC belongs to sector OBC of a circle with centre O.

It is evident from the diagram that OB=OC are radii of the circle with centre O. We have found in (i)  that OB=7.270. Therefore;

But we also need angle BOC to find length of arc BC.

It is evident from the diagram that;

It is given that OA is perpendicular to OC.

It is also given that angle AOB = 0.6;

Now we are in a position to find the length of arc BC.

Finally, we can find perimeter of the metal plate.

iii.

It is evident from the diagram that;

First we find area of triangle OAB.

Expression for the area of the triangle is;

For the given case;

We are given that  and from (ii) we have found that . Hence;

Next we need to find area of sector OBC.

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

We have found in (ii) that OB=OC are radii of the circle with centre O. We have found in (i)  that  OB=7.270.

From (ii) we also know that angle of circular sector OBC is;

Therefore;

Finally;