This isn't even funny how hard it is

Dividing both sides by $\pi$, $16x^2+12x+ 1= 182$Subtracting 182 from both sides, $16x^2+ 12x- 181= 0$This quadratic equation can be solved to find the value of $x$. Once $x$ is found, we can determine the diameter of the cylinder by substituting it into the expression $4x+2$. In summary, we can determine the diameter of a cylinder with a given surface area of $182\pi$ by solving the quadratic equation $16x^2+
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  • #2
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

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

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