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A typical chips and crepe packaging cone, for example, has V = 355 cm3.) What dimensions (height and radius) will minimize the cost of recycled paper to construct the cone?

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closed as off-topic by max_zorn, GNUSupporter 8964民主女神 地下教會, Ak19, Thomas Shelby, Lord Shark the Unknown Jun 16 at 5:50

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    $\begingroup$ For homework problems it is generally expected that some marginal effort will be spent on a solution... $\endgroup$ – copper.hat Jun 6 at 13:25
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    $\begingroup$ Make the entire cone of non-recycled paper. Guaranteed minimum. $\endgroup$ – Henning Makholm Jun 6 at 13:26
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  1. Presume cost of material is proportional to surface area. Thus seek to minimize surface area. Look up the formula for surface area of a cone (in terms of its height and radius).

  2. Write the surface area in terms of one variable (r or h). This requires a constraint relationship, which is the equation for volume of a cone. The volume is a constant so r can be written in terms of h, or vice versa, and substituted into the surface area equation to reduce it to a single variable (whichever variable is more convenient).

  3. Differentiate surface area with respect to the chosen variable, set the derivative equal to zero, and solve for the minimizing value of the variable (r or h).

  4. Plug the minimizing value of r (or h) into the constraint (volume) equation to find the value of the other variable.

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You will need the formula for the surface area, which is given by $$M=\pi r\sqrt{r^2+h^2}+\pi r^2$$ and the formula for the volume $$V=\frac{1}{3}\pi r^2h$$ so we have to optimize the function $$g(r)=\pi r\sqrt{r^2+\left(\frac{3V}{\pi r^2}\right)^2}+\pi r^2$$

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