Let r be the radius of cylinder and h be the height of cylinder. If the radius of a cylinder is increased by N%, then it becomes [tex] R=r+\dfrac{N}{100} r [/tex]. If its height increased by 2N%, then it becomes [tex] H=h+\dfrac{2N}{100} h [/tex].
The volume of initial cylinder is [tex] V_{initial}=\pi r^2h [/tex] and the volume of new cylinder is [tex] V_{new}=\pi R^2H=\pi \left(r+\dfrac{N}{100} r\right)\cdot \left(h+\dfrac{2N}{100}h\right)=\pi r^2h \left(1+\dfrac{N}{100} \right) \left(1+\dfrac{2N}{100} \right) [/tex].
The ratio between volumes is
[tex] \dfrac{V_{new}}{V_{initial}}=\dfrac{\pi r^2 h\left(1+\dfrac{N}{100} \right) \left(1+\dfrac{2N}{100} \right)}{\pi r^2 h} = \left(1+\dfrac{N}{100} \right) \left(1+\dfrac{2N}{100} \right) [/tex].
This means that volume increases [tex] \left(1+\dfrac{N}{100} \right) \left(1+\dfrac{2N}{100} \right) [/tex] times.
1. When N=20, substitute this value into previous expression:
[tex] \left(1+\dfrac{N}{100} \right) \left(1+\dfrac{2N}{100} \right)=\left(1+\dfrac{20}{100} \right) \left(1+\dfrac{40}{100} \right)=\left(1+\dfrac{1}{5} \right) \left(1+\dfrac{2}{5} \right)=\dfrac{6}{5} \cdot \dfrac{7}{5} =\dfrac{42}{25} =1.68 [/tex]. The coeeficient 1.68 in percent is 168% and this means that volume increases by 68%.
2. When the radius is decreased by 5%, then in first brackets you should subtract fraction and the height increased by 5%, then in secondt brackets you should add fraction. So,
[tex] \left(1-\dfrac{N}{100} \right) \left(1+\dfrac{N}{100} \right)=\left(1-\dfrac{5}{100} \right) \left(1+\dfrac{5}{100} \right) =\left(1-\dfrac{1}{20} \right) \left(1+\dfrac{1}{20} \right) =\dfrac{19}{20}\cdot \dfrac{21}{20}=\dfrac{399}{400}=0.9975[/tex].
The coeeficient 0.9975 in percent is 99.75% and this means that volume decreases by 0.25%.
