Question

g A manufacturer is making cylindrical cans that hold 300 cm3. The dimensions of the can are not mandated, so to save manufacturing costs, the manufacturer wants to make the can that uses the least material. What are the dimensions, to the nearest three decimal places, of the can that has the smallest surface area

Answer 1

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:

`\displaystyle V = \pi r^2h`

Substitute:

`\displaystyle (300) = \pi r^2 h`

Solve for h:

`\displaystyle (300)/(\pi r^2) = h`

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

`\displaystyle A = 2\pi r^2 + 2\pi rh`

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.

`\displaystyle \begin{aligned} A &= 2\pi r^2 + 2\pi r\left((300)/(\pi r^2)\right) \\ \\ &=2\pi r^2 + (600)/( r) \end{aligned}`

Find its derivative:

`\displaystyle A' = 4\pi r - (600)/(r^2)`

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}`

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}`

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.