The required equation of the surface area of the cylinder is [tex](60\pi x^2+192\pi x+128\pi)[/tex]
Solution:
Let "r" be the radius of cylinder and "h" be the height.
Formula for surface area is:
[tex]S= 2r^2\pi + 2r\pi h[/tex]
where [tex]2r^2\pi[/tex] is surface of two bases of cylinder and [tex]2r\pi h[/tex] is surface of mantle.
From the given, the height of the gift box is three times the radius, i.e [tex]h=3r=3\times(3x+4)\rightarrow h=9x+12[/tex]
On substituting the given values in the formula we get,
[tex]S = 2\times\pi[(3x+4)^2 + (9x+12)\times(3x+4)][/tex]
[tex]\Rightarrow S=2\pi[(3x^2)+2\times3x\times4+4^2]+[(9x\times3x)+12\times3x+36x+48][/tex]
[tex]\Rightarrow S=2\pi[9x^2+24x+16]+[21x^2+36x+36x+48][/tex]
[tex]\Rightarrow S=2\pi[9x^2+24x+16+21x^2+72x+48]\rightarrow2\pi[30x^2+96x+64][/tex]
On multiplying [tex]2\pi[/tex] inside the brackets we get,
[tex]\Rightarrow S=[60\pi x^2+192\pi x+128\pi][/tex]
Hence, the required equation is [tex](60\pi x^2+192\pi x+128\pi)[/tex]
