Question

A company makes paper labels for cans of soup. As shown below, each can is in the shape of a cylinder with a height of 7cm and a diameter of 8cm. The paper label is wrapped around the can and covers only the side of the can (not the top or bottom). If the company makes 14 labels, how many square centimeters of paper will be needed? Use 3.14 for pi, and do not round your answer.

Answer 1

To calculate the amount of paper needed for the labels, we will determine the surface area of the side of one cylindrical can and then multiply by the number of labels.

The formula for the lateral surface area (side area) of a cylinder is given by:

\[
\text{Lateral Surface Area} = 2 \pi r h
\]

Where:
- \( r \) is the radius of the cylinder
- \( h \) is the height of the cylinder
- \( \pi \) is approximately 3.14

Given:
- The diameter of the can is 8 cm, so the radius \( r \) is:

\[
r = \frac{\text{diameter}}{2} = \frac{8 \, \text{cm}}{2} = 4 \, \text{cm}
\]

- The height \( h \) of the can is:

\[
h = 7 \, \text{cm}
\]

Now, substituting the values into the formula:

\[
\text{Lateral Surface Area} = 2 \times 3.14 \times 4 \, \text{cm} \times 7 \, \text{cm}
\]

Calculating step-by-step:

1. Calculate \( 2 \times 3.14 = 6.28 \).
2. Calculate \( 6.28 \times 4 = 25.12 \).
3. Finally, calculate \( 25.12 \times 7 = 175.84 \).

Therefore, the lateral surface area of one can is:

\[
\text{Lateral Surface Area} = 175.84 \, \text{cm}^2
\]

Since the company makes 14 labels, we multiply the area of one label by 14:

\[
\text{Total area for 14 labels} = 14 \times 175.84 \, \text{cm}^2
\]

Calculating this gives:

\[
14 \times 175.84 = 2461.76 \, \text{cm}^2
\]

Thus, the total amount of paper needed for 14 labels is:

\[
\boxed{2461.76} \, \text{cm}^2
\]