A pringles can have a diameter of 3 inches and a height of 8.5 inches. About how much area does the label cover? Round answer to nearest hundredth of a square inch.
The Correct Answer and Explanation is :
To determine how much area the label on a Pringles can covers, we need to calculate the surface area of the cylindrical can. The label usually covers the lateral surface area (side) of the cylinder, which can be calculated using the formula for the lateral surface area of a cylinder:
[
\text{Lateral Surface Area} = 2\pi r h
]
Where:
- (r) is the radius of the base (half of the diameter),
- (h) is the height of the cylinder,
- (\pi) is approximately 3.1416.
Step 1: Identify the dimensions of the can
- Diameter of the can = 3 inches
- Radius (r = \frac{3}{2} = 1.5) inches
- Height (h = 8.5) inches
Step 2: Plug the values into the formula
[
\text{Lateral Surface Area} = 2\pi(1.5)(8.5)
]
[
\text{Lateral Surface Area} = 2 \times 3.1416 \times 1.5 \times 8.5
]
[
\text{Lateral Surface Area} = 2 \times 3.1416 \times 12.75
]
[
\text{Lateral Surface Area} = 80.1072 \, \text{square inches}
]
Step 3: Round the result
Rounding (80.1072) to the nearest hundredth gives:
[
\text{Lateral Surface Area} \approx 80.11 \, \text{square inches}
]
Explanation:
This result represents the area that the label on the Pringles can covers. The label wraps around the cylindrical surface, which is why we calculate the lateral surface area instead of the area of just the top or bottom of the can. The formula for the lateral surface area of a cylinder is derived from the fact that the side of the cylinder can be “unwrapped” into a rectangle, where the height of the rectangle is the height of the cylinder, and the width of the rectangle is the circumference of the circular base. The circumference of the circle is given by (2\pi r), and the area of the rectangle (lateral surface area) is the product of the circumference and height.