Let r cm be the radius of the base and h cm be the height of the cylinder. Then, total surface area of cylinder {tex}= 2\\pi r(r + h){/tex}Curved surface area of cylinder {tex}= 2\\pi rh{/tex}We have,Curved surface area = {tex}\\frac{1}{3}{/tex}(Total surface area) = {tex}\\frac{1}{3}{/tex}{tex}\\times{/tex}\xa0{tex}462 cm^2 = 154 cm^2\xa0{/tex}{tex}\\Rightarrow{/tex}{tex}2\\pi rh = 154{/tex}Also, {tex}2\\pi rh + 2\\pi r^2 = 462 {/tex}{tex}\\Rightarrow{/tex}154 + 2{tex}\\pi{/tex}{tex}r^2 = 462{/tex}{tex}\\Rightarrow{/tex}2{tex}\\pi{/tex}{tex}r^2 = 462 – 154 = 308 cm^2{/tex}2 {tex}\\times{/tex}{tex}\\frac{22}{7}{/tex}{tex}\\times{/tex}{tex}r^2 = 308\xa0{/tex}{tex}\\Rightarrow{/tex}r2 ={tex}\\frac{308 \\times 7}{2 \\times22}{/tex} = 72{tex}\\Rightarrow{/tex}\xa0{tex}r = 7 cm{/tex}Again 2{tex}\\pi{/tex}rh = 154 {tex}\\Rightarrow{/tex}\xa02{tex}\\times{/tex}{tex}\\frac{22}{7}{/tex}{tex}\\times{/tex}7{tex}\\times{/tex}{tex}h = 154{/tex}{tex}\\Rightarrow{/tex}\xa0h = {tex}\\frac{154}{2 \\times22}{/tex}={tex}\\frac{7}{2}{/tex}cmVolume of the cylinder = {tex}\\pi{/tex}r2h = {tex}\\frac{22}{7}{/tex}{tex}\\times{/tex}\xa07 {tex}\\times{/tex}\xa07 {tex}\\times{/tex}\xa0{tex}\\frac{7}{2}{/tex}\xa0{tex}= 539 cm^3{/tex}