To answer this question, let's calculate the surface area of the can for both the label and the entire can.

Given Data

• Diameter of the can $= 6$ cm, so radius $r = \frac{6}{2} = 3$ cm.
• Height of the can $h = 10$ cm.

1. Surface Area for the Label

The label typically covers only the curved surface area of the can. The formula for the lateral (curved) surface area of a cylinder is: $\text{Lateral Surface Area} = 2 \pi r h$ Substituting in the values: $= 2 \pi (3) (10) = 60 \pi \approx 188.4 \text{ square centimeters}$

So, the can requires approximately 188.4 square centimeters of paper to create the label.

2. Total Surface Area for the Metal Can

To find the total surface area (metal needed), we need to consider both the curved surface and the two circular bases (top and bottom). The formula for the total surface area of a cylinder is: $\text{Total Surface Area} = 2 \pi r h + 2 \pi r^2$ Substituting in the values: $= 2 \pi (3) (10) + 2 \pi (3)^2$ $= 60 \pi + 18 \pi = 78 \pi \approx 245.0 \text{ square centimeters}$

So, the can requires approximately 245.0 square centimeters of metal to make the can.

Final Answer

The can of peaches requires 188.4 square centimeters of paper for the label and 245.0 square centimeters of metal to make the can.

Would you like further details or have any questions?

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Related Questions:

1. How would these calculations change if the height of the can increased?
2. How is the surface area of a can affected if the radius is doubled?
3. What is the difference between lateral surface area and total surface area?
4. How could these surface area calculations be applied to a real-world packaging scenario?
5. Why is the curved surface area used for the label rather than the total surface area?

Tip:

When working with cylinders, always identify which parts (curved or total) are required to avoid overestimating or underestimating material needs.