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.

Question

Philoid · Accepted Answer

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 = 90^o

Now, in right triangle OAP, we have

Also, in right triangle OAP we have:

So, ∠ AOB = 60^o + 60^o

= 120^o (i)

∴ Reflex ∠ AOB = 360^o – 120^o

= 240^o

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 cm}²