- #1
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The discussion focuses on calculating the total surface area of a cylinder using the formula A = 2πr² + 2πrh. Given the radius r = (4x + 2)/2 and height h = 2x, the equation is set equal to 182π to solve for x. After substituting r and h into the surface area formula, the resulting quadratic equation 16x² + 12x - 181 = 0 is derived. The participants discuss solving this quadratic to find the value of x, which is then used to determine the diameter of the cylinder. The calculations emphasize the importance of correctly applying geometric formulas in problem-solving.
- #2
skeeter
- 1,103
- 1
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
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
- #3
HOI
- 921
- 2
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.
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
skeeter
- 1,103
- 1
$2\pi (2x+1)^2 + \pi(8x^3+4x) = \pi(8x^2+8x + {\color{red}2} +8x^2+4x)$