Answer:
2. b : l
3. 20cm
4. 49 ![cm^{2}](https://tex.z-dn.net/?f=cm%5E%7B2%7D)
5. ![(2\pi+1):2\pi](https://tex.z-dn.net/?f=%282%5Cpi%2B1%29%3A2%5Cpi)
Step-by-step explanation:
<u>Solution 2:</u>
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 = \dfrac{b}{2\pi}](https://tex.z-dn.net/?f=2%5Cpi%20r%20%3D%20b%5C%5C%5CRightarrow%20r%20%3D%20%5Cdfrac%7Bb%7D%7B2%5Cpi%7D)
Volume of a cylinder is given as:
![V = \pi r^{2} h](https://tex.z-dn.net/?f=V%20%3D%20%5Cpi%20r%5E%7B2%7D%20h)
Putting the values:
![V_1 = \pi (\dfrac{b}{2\pi})^2 l\\\Rightarrow V_1 = (\dfrac{b^2}{4\pi}) l](https://tex.z-dn.net/?f=V_1%20%3D%20%5Cpi%20%28%5Cdfrac%7Bb%7D%7B2%5Cpi%7D%29%5E2%20l%5C%5C%5CRightarrow%20V_1%20%3D%20%20%28%5Cdfrac%7Bb%5E2%7D%7B4%5Cpi%7D%29%20l)
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 = \dfrac{l}{2\pi}](https://tex.z-dn.net/?f=2%5Cpi%20r%20%3D%20l%5C%5C%5CRightarrow%20r%20%3D%20%5Cdfrac%7Bl%7D%7B2%5Cpi%7D)
Volume of a cylinder is given as:
![V = \pi r^{2} h](https://tex.z-dn.net/?f=V%20%3D%20%5Cpi%20r%5E%7B2%7D%20h)
Putting the values:
![V_2 = \pi (\dfrac{l}{2\pi})^2 b\\\Rightarrow V_2 = (\dfrac{l^2}{4\pi}) b](https://tex.z-dn.net/?f=V_2%20%3D%20%5Cpi%20%28%5Cdfrac%7Bl%7D%7B2%5Cpi%7D%29%5E2%20b%5C%5C%5CRightarrow%20V_2%20%3D%20%20%28%5Cdfrac%7Bl%5E2%7D%7B4%5Cpi%7D%29%20b)
Taking ratio:
![V_1:V_2 = \dfrac{(\dfrac{b^2}{4\pi}) l}{(\dfrac{l^2}{4\pi}) b} = b:l](https://tex.z-dn.net/?f=V_1%3AV_2%20%3D%20%5Cdfrac%7B%28%5Cdfrac%7Bb%5E2%7D%7B4%5Cpi%7D%29%20l%7D%7B%28%5Cdfrac%7Bl%5E2%7D%7B4%5Cpi%7D%29%20b%7D%20%3D%20b%3Al)
Solution 3:
Rectangle is rolled along its length to make a cylinder, so height will be equal to its length.
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.
= 7 cm
Curved surface area of a cylinder:
![CSA = 2\pi rh](https://tex.z-dn.net/?f=CSA%20%3D%202%5Cpi%20rh)
Putting the above values:
CSA = 7
7 = 49 ![cm^{2}](https://tex.z-dn.net/?f=cm%5E%7B2%7D)
Solution 5:
As calculated in above step:
CSA =
7
7 = 49 ![cm^{2}](https://tex.z-dn.net/?f=cm%5E%7B2%7D)
Total surface area = ![2\pi r^{2} + 2\pi r h](https://tex.z-dn.net/?f=2%5Cpi%20r%5E%7B2%7D%20%2B%202%5Cpi%20r%20h)
Calculating value of r:
= 7 cm
![\Rightarrow 2 \pi r = 7\\\Rightarrow r = \dfrac{7}{2\pi}](https://tex.z-dn.net/?f=%5CRightarrow%202%20%20%5Cpi%20r%20%3D%207%5C%5C%5CRightarrow%20r%20%3D%20%5Cdfrac%7B7%7D%7B2%5Cpi%7D)
Total surface area =
![2\pi (\dfrac{7}{2\pi})^{2} + 49\\\Rightarrow \dfrac{49}{2\pi}+49\\\Rightarrow 49(\dfrac{1}{2\pi}+1) cm^2](https://tex.z-dn.net/?f=2%5Cpi%20%28%5Cdfrac%7B7%7D%7B2%5Cpi%7D%29%5E%7B2%7D%20%2B%2049%5C%5C%5CRightarrow%20%5Cdfrac%7B49%7D%7B2%5Cpi%7D%2B49%5C%5C%5CRightarrow%2049%28%5Cdfrac%7B1%7D%7B2%5Cpi%7D%2B1%29%20cm%5E2)
Ratio of TSA: CSA is
![49(\dfrac{1}{2\pi}+1) cm^2 : 49 cm^2\\\Rightarrow (\dfrac{1}{2\pi}+1):1\\\Rightarrow (2\pi+1): 2\pi](https://tex.z-dn.net/?f=49%28%5Cdfrac%7B1%7D%7B2%5Cpi%7D%2B1%29%20cm%5E2%20%3A%2049%20cm%5E2%5C%5C%5CRightarrow%20%28%5Cdfrac%7B1%7D%7B2%5Cpi%7D%2B1%29%3A1%5C%5C%5CRightarrow%20%282%5Cpi%2B1%29%3A%202%5Cpi)