Question

The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions to move once over a level a playground. Find the area of the playground in \[{{m}^{2}}\].
(a) \[1584\,{{m}^{2}}\]
(b) \[1284\,{{m}^{2}}\]
(c) \[1554\,{{m}^{2}}\]
(d) none of the above

Answer 1

Hint: The roller is in the shape of a right circular cylinder. To find the area that the roller covers the playground in five revolutions, we need to find the lateral area of the roller. The lateral area of the roller is equal to the area of its curved surface.

Complete step-by-step solution -
The formula for finding the lateral area of a right circular cylinder is: \[\text{L=}2\pi rh......(1)\] where L is the lateral surface area of the cylinder, r is the radius of the cylinder and h is the cylinder’s height.
Since the roller is positioned horizontal as it rolls along the playground, then the length of the roller is actually its height.

Now in the question diameter is mentioned and we need to find a radius which is half of diameter.
Diameter of the roller \[=d=84\,cm\]
Radius of the roller \[=r=\dfrac{d}{2}=\dfrac{84}{2}=42\,cm\]
Height of the roller \[=h=120\,\,cm\]
Now putting value of r and h in equation (1) to get the lateral area of the roller we get,
 \[\text{L=}2\times \dfrac{22}{7}\times 42\times 120=31680\,c{{m}^{2}}......(2)\]
 \[\text{A}=\text{n }\!\!\times\!\!\text{ L}.....(3)\] where A is the area covered by the roller, n is the number of revolutions and L is the lateral area of the roller.
In question the given number of revolutions is 500 that is \[\text{n}=500\]. Now substituting this value of n and the value of L from equation (2) in equation (3) we get,
 \[\text{A}=500\text{ }\!\!\times\!\!\text{ 31680=15840000 c}{{m}^{2}}\]
Converting this area in \[{{m}^{2}}\] we get \[\text{A}=\dfrac{15840000}{10000}=1584\,{{m}^{2}}\]
Hence the area covered by the roller in the 500 revolution is \[1584\,{{m}^{2}}\]. So the correct answer is option (a).

Note: The key here is to understand that the roller is nothing but a cylinder. Knowing the formula of lateral area is important and radius is always half of the diameter. We in a hurry may substitute diameter which is 84 cm in place of radius which is 42 cm. We need to be careful with the units too because if a unit is missing some marks would be deducted by the examiner.