The question in the image asks:

"Wie viel cm² Blech braucht man für eine Dose (ohne Verschnitt)?"

In English: "How many square centimeters of metal sheet are needed for a can (without cutting losses)?"

We are given the dimensions of a cylindrical can:

• Height (h) = 11.5 cm
• Diameter = 10 cm, hence the radius (r) = 5 cm (since $r = \frac{\text{diameter}}{2}$).

To calculate the total surface area of the can, we need to sum:

1. The area of the cylindrical side (lateral surface area).
2. The area of the top and bottom (which are circles).

Step-by-step:

1. Lateral surface area (A₁) of a cylinder: $A₁ = 2 \pi r h$
2. Top and bottom area (A₂) (two circles): $A₂ = 2 \pi r^2$
3. Total surface area (without cutting losses): $A_{\text{total}} = A₁ + A₂ = 2 \pi r h + 2 \pi r^2$ Now, let me compute this.The total surface area of the metal sheet needed to make the can, without any cutting losses, is approximately 518.36 cm².

Would you like further details or clarifications?

Here are 5 related questions to expand on this:

1. How would the calculation change if the can had a lid of different thickness?
2. What is the volume of the can based on the given dimensions?
3. How would the total area change if the height of the can increased by 10%?
4. How is the surface area formula derived for other shapes like a cone or a sphere?
5. What is the impact of rounding the values of π or dimensions on such surface area calculations?

Tip: When calculating surface areas or volumes, always ensure the units of measurement are consistent to avoid errors.