- #1
This isn't even funny how hard it is
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Discussion Overview
The discussion revolves around solving a problem related to the total surface area of a cylinder, specifically involving the substitution of variables for radius and height in terms of a variable \(x\). Participants are engaged in mathematical reasoning and problem-solving related to this geometry topic.
Discussion Character
- Mathematical reasoning
- Homework-related
Main Points Raised
- One participant states the formula for the total surface area of a cylinder as \(A = 2\pi r^2 + 2\pi rh\) and provides expressions for \(r\) and \(h\) in terms of \(x\).
- Another participant simplifies the expressions for the areas of the top, bottom, and side of the cylinder, leading to a quadratic equation \(16x^2 + 12x - 181 = 0\) after substituting and combining terms.
- A later reply appears to contain a potential error in the expression for the side area, introducing a term that may not align with previous calculations.
Areas of Agreement / Disagreement
Participants are engaged in a collaborative problem-solving process, but there are indications of potential discrepancies in the calculations, particularly regarding the expression for the side area. The discussion remains unresolved as participants have not reached a consensus on the correctness of the calculations.
Contextual Notes
There are unresolved mathematical steps, particularly concerning the simplification of the side area and the introduction of terms that may not be consistent with the earlier calculations. The discussion relies on the definitions of radius and height as functions of \(x\).
- #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
$2\pi (2x+1)^2 + \pi(8x^3+4x) = \pi(8x^2+8x + {\color{red}2} +8x^2+4x)$
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