Q43 of 45 Page 12

An elastic belt is placed around the rim of a pulley of radius 5 cm. (Fig. 10) From one-point C on the belt, the elastic belt is pulled directly away from the center O of the pulley until it is at P, 10 cm from the point O. Find the length of belt that is still in contact with the pulley. Also find the shaded area.

(CBSE 2016)

On the basis of question the figure can be redrawn as follows:


We know that, tangent of a circle is perpendicular to the radius of the circle through the point of contact


∴∠ OAP = 90o


Now, in right triangle OAP, we have





Also, in right triangle OAP we have:






So, ∠ AOB = 60o + 60o


= 120o (i)


∴ Reflex ∠ AOB = 360o – 120o


= 240o


We know that, length of the belt which is in contact with the pulley = ADB


= Length of major arc





= 20.93̅ cm


Now, in right angled triangle OAP we have:





We know that,





∴


We know that,




= 26.17 cm2 (Approx.)


∴ Area of shaded region = Area of triangle OAP + Area of triangle OBP – Area of minor sector OABC




= 25 × 1.73 – 26.17


= 17.08 cm2


Hence, area of the shaded region will be 17.08 cm2

More from this chapter

All 45 →
41

A child makes a poster on a chart paper drawing a square ABCD of side 14cm. She draws four circles with centre A,B,C and D in which she suggests different ways to save energy. The circles are drawn in such away that each circle touches externally two of the three remaining circles (Fig.15.74). In the shaded region she writes a message ‘Save Energy’. Find the perimeter and area of the shaded region.

(CBSE 2015)

 42

In fig. 7, are shown two arcs PAQ and PBQ. Arc PAQ is a part of circle with center O and radius OP while arc PBQ is a semi-circle drawn on PQ as diameter with center M. If OP = PQ = 10 cm, show that area of shaded region is.

(CBSE 2016)

 44

In the given figure, ABCD is a rectangle of dimensions 21 cm × 14 cm. A semicircle is drawn with BC as diameter. Find the area and the perimeter of the shaded region in the figure.

(CBSE 2017)

 45

In the given figure, the side of square is 28 cm and radius of each circle is half of the length of the side of the square where O and O′ are centers of the circles. Find the area of shaded region.

(CBSE 2017)