This isn't even funny how hard it is

[gilamonster]

https://assets.learnosity.com/organisations/154/a4840bb5-1a32-4997-8d1f-28992a862a2b.PNG

 

[skeeter]

total surface area of a cylinder, $A = 2\pi r^2 + 2\pi rh$

from the sketch, $r = \dfrac{4x+2}{2}$ and $h=2x$

substitute for $r$ and $h$ and set equal to $182\pi$, solve for $x$, then determine the diameter

 

[HOI]

That's pretty straight forward. The diameter is 4x+ 2 so the radius is 2x+ 1. The top and bottom each have area $\pi r^2= \pi (2x+ 1)^2$. That totals $2\pi(2x+ 1)^2$. For the side, imagine cutting down the side and "unrolling" it. You get a rectangle with width 2x and length equal to the circumferce of the circles $2\pi r= \pi(4x+ 2)$, That area is $\pi(8x^2+ 4x)$ so the total area is $2\pi(2x+ 1)^2+ \pi(8x^2+4x)= \pi(8x^2+ 8x+ 1+ 8x^2+ 4x)= \pi(16x^2+ 12c+ 1)= 182\pi$.

Dividing both sides by $\pi$, $16x^2+12x+ 1= 182$. Subtracting 182 from both sides, $16x^2+ 12x- 181= 0$. Solve that quadratic equation for x.

 

[skeeter]

$2\pi (2x+1)^2 + \pi(8x^3+4x) = \pi(8x^2+8x + {\color{red}2} +8x^2+4x)$