To solve this problem, we are tasked with finding a cylindrical design that maintains the same volume (178 cm³) but minimizes the surface area. This will help reduce the metal used in production.

Step 1: Define the problem mathematically

1. Volume of a cylinder: $V = \pi r^2 h$ Given $V = 178 \, \text{cm}^3$, the height $h$ can be expressed as: $h = \frac{V}{\pi r^2}$

2. Surface area of a cylinder (to minimize): $S = 2\pi r^2 + 2\pi r h$ Here, the first term $2\pi r^2$ represents the area of the two circular ends, and the second term $2\pi r h$ represents the lateral surface area.

Step 2: Substitute $h$ into the surface area equation

From the volume equation, substitute $h = \frac{178}{\pi r^2}$ into the surface area formula: $S = 2\pi r^2 + 2\pi r \left( \frac{178}{\pi r^2} \right)$

Simplify the second term: $S = 2\pi r^2 + \frac{356}{r}$

Step 3: Minimize the surface area

To find the value of $r$ that minimizes $S$, we take the derivative of $S$ with respect to $r$, set it equal to zero, and solve for $r$:

1. Derivative of $S$: $\frac{dS}{dr} = 4\pi r - \frac{356}{r^2}$

2. Set $\frac{dS}{dr} = 0$: $4\pi r = \frac{356}{r^2}$

3. Solve for $r$: Multiply through by $r^2$: $4\pi r^3 = 356$ Solve for $r^3$: $r^3 = \frac{356}{4\pi} \quad \text{or} \quad r^3 \approx \frac{356}{12.566} \approx 28.33$ Take the cube root: $r \approx \sqrt[3]{28.33} \approx 3.04 \, \text{cm}$

Step 4: Find the corresponding height $h$

Using $h = \frac{178}{\pi r^2}$ and $r \approx 3.04$: $h = \frac{178}{\pi (3.04)^2}$ $h \approx \frac{178}{29.07} \approx 6.12 \, \text{cm}$

Final Answer:

• The optimal radius is approximately 3.04 cm.
• The corresponding height is approximately 6.12 cm.

This design maintains the volume of 178 cm³ but minimizes the surface area, which reduces the material used.

Do you need further clarifications, or would you like me to show detailed steps using graphing technology?

Here are 5 related questions to expand on this:

1. How do you derive the surface area formula for a cylinder?
2. What is the significance of minimizing surface area in real-world applications?
3. How would the result change if the volume were doubled?
4. Can we apply a similar optimization to other shapes, like spheres or cones?
5. How does calculus help in solving optimization problems?

Tip: In optimization problems, always express the target function in terms of a single variable before differentiating.