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How do you calculate the surface area to volume ratio of a cylinder?

Answer
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Hint: In the given question, we have been asked to find the ratio of surface area of cylinder to the volume of cylinder. In order to find the ratio, we will need to simply divide the surface area of the cylinder by the volume of the cylinder. The formula of surface area of cylinder is given by 2πr2+2πrh and the formula of the volume of the cylinder is given byπr2h.

Complete step-by-step solution:
We have the given 3-dimensional shape i.e. cylinder
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Surface area of the cylinder is given by the formula;
Surface area of cylinder = area of the circular top + area of the circular bottom + area of the side
As we know that,
Area of circle = πr2
Area of the side i.e. in rectangular form when we open it = length×breadth
As we can see that the length is equal to the circumference of a circle.
Thus,
Area of the side i.e. in rectangular form when we open it = length×breadth=2πr×h
Therefore,
Surface area of cylinder = area of the circular top + area of the circular bottom + area of the side
πr2+πr2+2πrh
2πr2+2πrh
Surface area of the cylinder = 2πr2+2πrh
Now,
The volume of the cylinder is given by the formula;
Volume of a cylinder = πr2h
So,
We have given the ratio of surface area of the cylinder to the volume of the cylinder;
surface area of cylindervolume of cylinder=2πr2+2πrhπr2h
Here, ‘r’ is the radius of the cylinder and ‘h’ is the height of the cylinder.
Let take an example;
Suppose we have a cylinder of radius 6cm and height 12cm.
Then,
Surface area of the cylinder = 2πr2+2πrh=2π(6)2+2π×6×12=72π+144π=216π
Therefore, the surface area of the cylinder is216π.
Volume of the cylinder = πr2h=π(6)2×12=π×36×12=432π
Therefore, the volume of the cylinder is432π.
Now,
The ratio of surface area of the cylinder to the volume of the cylinder;
surface area of cylindervolume of cylinder=2πr2+2πrhπr2h=216π432π=12

Therefore the ratio of surface area of the cylinder to the volume of the cylinder is 12.

Note: A cylinder is a three dimensional figure i.e. a figure that has length, breadth and the height. The key note is that the units for surface area would be cm2orm2, whereas the units for the volume of the 3-D figure would becm3 or m3. To solve the question easily, do not put the value of πwhile calculating the ratio of surface area to volume as it will cancel out in the simplification and you will get the required ratio in an easy way as it will not complicate the answer.
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