Question

Q2.What is the ratio of volumes of the two cylinders formed by rolling a sheet of dimensions lxb in two different ways? Q3.What is the height of a right circular cylinder made from a rectangular sheet of length 20cm and breadth 15cm if it’s rolled along its length? Q4.What is the curved surface area of a cylinder obtained by rolling a square along its 7cm long side? Q5.What is the ratio of the total surface area and the curved surface area of the cylinder?

Answer 1

2. b : l

3. 20cm

4. 49
`cm^(2)`

5.
`(2\pi+1):2\pi`

Explanation:

Solution 2:

Let cylinder is rolled along 'l':

Height of cylinder , h = length of rectangle = l

Perimeter of base = b

Let 'r' be the radius of cylinder's base:

`2\pi r = b\\\Rightarrow r = (b)/(2\pi)`

Volume of a cylinder is given as:

`V = \pi r^(2) h`

Putting the values:

`V_1 = \pi ((b)/(2\pi))^2 l\\\Rightarrow V_1 = ((b^2)/(4\pi)) l`

Let cylinder is rolled along 'b':

Height of cylinder , h = length of rectangle = b

Perimeter of base = l

Let 'r' be the radius of cylinder's base:

`2\pi r = l\\\Rightarrow r = (l)/(2\pi)`

Volume of a cylinder is given as:

`V = \pi r^(2) h`

Putting the values:

`V_2 = \pi ((l)/(2\pi))^2 b\\\Rightarrow V_2 = ((l^2)/(4\pi)) b`

Taking ratio:

`V_1:V_2 = (((b^2)/(4\pi)) l)/(((l^2)/(4\pi)) b) = b:l`

Solution 3:

Rectangle is rolled along its length to make a cylinder, so height will be equal to its length.

`\therefore` height of cylinder = 20 cm

Solution 4:

Side of square = 7 cm

Height of cylinder =Side of square = 7 cm

7 cm will be the circumference of the circle.

i.e.
`2\pi r` = 7 cm

Curved surface area of a cylinder:

`CSA = 2\pi rh`

Putting the above values:

CSA = 7
`*` 7 = 49
`cm^(2)`

Solution 5:

As calculated in above step:

CSA =
`2\pi rh =` 7
`*` 7 = 49
`cm^(2)`

Total surface area =
`2\pi r^(2) + 2\pi r h`

Calculating value of r:

`2\pi r` = 7 cm

`\Rightarrow 2 \pi r = 7\\\Rightarrow r = (7)/(2\pi)`

Total surface area =

`2\pi ((7)/(2\pi))^(2) + 49\\\Rightarrow (49)/(2\pi)+49\\\Rightarrow 49((1)/(2\pi)+1) cm^2`

Ratio of TSA: CSA is

`49((1)/(2\pi)+1) cm^2 : 49 cm^2\\\Rightarrow ((1)/(2\pi)+1):1\\\Rightarrow (2\pi+1): 2\pi`