Answer:
The dimensions that minimize the cost of materials for the cylinders have radii of about 3.628 cm and heights of about 7.256 cm.
Explanation:
A cylindrical can holds 300 cubic centimeters, and we want to find the dimensions that minimize the cost for materials: that is, the dimensions that minimize the surface area.
Recall that the volume for a cylinder is given by:

Substitute:

Solve for h:

Recall that the surface area of a cylinder is given by:

We want to minimize this equation. To do so, we can find its critical points, since extrema (minima and maxima) occur at critical points.
First, substitute for h.

Find its derivative:

Solve for its zero(s):
![\displaystyle \begin{aligned} (0) &= 4\pi r - (600)/(r^2) \\ \\ 4\pi r - (600)/(r^2) &= 0 \\ \\ 4\pi r^3 - 600 &= 0 \\ \\ \pi r^3 &= 150 \\ \\ r &= \sqrt[3]{(150)/(\pi)} \approx 3.628\text{ cm}\end{aligned}](https://img.qammunity.org/2022/formulas/mathematics/college/ctyhevsv32s976f36f9nc18hd8jb07it5a.png)
Hence, the radius that minimizes the surface area will be about 3.628 centimeters.
Then the height will be:
![\displaystyle \begin{aligned} h&= \frac{300}{\pi\left( \sqrt[3]{(150)/(\pi)}\right)^2} \\ \\ &= \frac{60}{\pi \sqrt[3]{(180)/(\pi^2)}}\approx 7.25 6\text{ cm} \end{aligned}](https://img.qammunity.org/2022/formulas/mathematics/college/joum1t5w7pprb9puzp6fvbmjvt0m2bf7ug.png)
In conclusion, the dimensions that minimize the cost of materials for the cylinders have radii of about 3.628 cm and heights of about 7.256 cm.