An open-top cylindrical container is to have a volume $343 \mathrm{~cm}^{3}$. What dimensions (radius and height) will minimize the surface area?
The radius of the can is about $\mathrm{cm}$ and its height is about $\mathrm{cm}$.
(Do not round until the final answer. Then round to two decimal places as needed.)
Answer
Final Answer: The radius of the can that minimizes the surface area is approximately \(\boxed{4.78 \mathrm{~cm}}\) and its height is also approximately \(\boxed{4.78 \mathrm{~cm}}\).
Step 1 ：The volume of a cylinder is given by the formula \(V = \pi r^2 h\) where \(r\) is the radius and \(h\) is the height. We are given that the volume is \(343 \mathrm{~cm}^{3}\), so we can express the height in terms of the radius as \(h = \frac{V}{\pi r^2}\).
Step 2 ：The surface area of an open-top cylinder is given by the formula \(A = \pi r^2 + 2\pi rh\). Substituting \(h\) from the volume equation into the surface area equation, we get \(A = \pi r^2 + \frac{2V}{r}\).
Step 3 ：To minimize the surface area, we need to find the derivative of \(A\) with respect to \(r\), set it equal to zero, and solve for \(r\). This will give us the radius that minimizes the surface area. We can then substitute this radius back into the height equation to find the corresponding height.
Step 4 ：The derivative of \(A\) with respect to \(r\) is \(2\pi r - \frac{686}{r^2}\). Setting this equal to zero and solving for \(r\) gives \(r \approx 4.78 \mathrm{~cm}\).
Step 5 ：Substituting \(r \approx 4.78 \mathrm{~cm}\) back into the height equation gives \(h \approx 4.78 \mathrm{~cm}\).
Step 6 ：Final Answer: The radius of the can that minimizes the surface area is approximately \(\boxed{4.78 \mathrm{~cm}}\) and its height is also approximately \(\boxed{4.78 \mathrm{~cm}}\).
