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Answer :
- Determine the radius of the soda can: $r = 3.25 / 2 = 1.625$ inches.
- Calculate the area of the two circular ends: $2 \times 3.14 \times (1.625)^2$.
- Calculate the lateral surface area: $2 \times 3.14 \times 1.625 \times 8$.
- Add the areas and round to the nearest tenth: The surface area is approximately $\boxed{98.2}$ square inches.
### Explanation
1. Problem Analysis
First, let's analyze the problem. We need to find the amount of aluminum required to make a soda can, which is essentially the surface area of a cylinder. The can has a diameter of $3 frac{1}{4}$ inches (3.25 inches) and a height of 8 inches. We'll use 3.14 as an approximation for $\pi$.
2. Finding the Radius
The surface area of a cylinder is given by the formula: $2 \pi r^2 + 2 \pi r h$, where $r$ is the radius and $h$ is the height. The radius is half of the diameter, so $r = 3.25 / 2 = 1.625$ inches.
3. Area of Circular Ends
Now, let's calculate the area of the two circular ends: $2 \pi r^2 = 2 * 3.14 * (1.625)^2$.
4. Lateral Surface Area
Next, we calculate the lateral surface area: $2 \pi r h = 2 * 3.14 * 1.625 * 8$.
5. Total Surface Area
Adding these two values together, we get the total surface area: $2 * 3.14 * (1.625)^2 + 2 * 3.14 * 1.625 * 8$.
6. Final Calculation
Calculating this gives us approximately 98.2 square inches. Therefore, the amount of aluminum required to make the soda can is approximately 98.2 square inches.
7. Final Answer
The amount of aluminum required to make the soda can is approximately $\boxed{98.2}$ square inches.
### Examples
Understanding surface area is crucial in manufacturing. For example, when designing packaging for food or beverages, knowing the surface area helps determine the amount of material needed, reducing waste and cost. This also applies to architecture, where calculating the surface area of walls and roofs helps estimate the amount of paint or roofing material required. Surface area calculations are also used in thermal engineering to estimate heat transfer from objects.
- Calculate the area of the two circular ends: $2 \times 3.14 \times (1.625)^2$.
- Calculate the lateral surface area: $2 \times 3.14 \times 1.625 \times 8$.
- Add the areas and round to the nearest tenth: The surface area is approximately $\boxed{98.2}$ square inches.
### Explanation
1. Problem Analysis
First, let's analyze the problem. We need to find the amount of aluminum required to make a soda can, which is essentially the surface area of a cylinder. The can has a diameter of $3 frac{1}{4}$ inches (3.25 inches) and a height of 8 inches. We'll use 3.14 as an approximation for $\pi$.
2. Finding the Radius
The surface area of a cylinder is given by the formula: $2 \pi r^2 + 2 \pi r h$, where $r$ is the radius and $h$ is the height. The radius is half of the diameter, so $r = 3.25 / 2 = 1.625$ inches.
3. Area of Circular Ends
Now, let's calculate the area of the two circular ends: $2 \pi r^2 = 2 * 3.14 * (1.625)^2$.
4. Lateral Surface Area
Next, we calculate the lateral surface area: $2 \pi r h = 2 * 3.14 * 1.625 * 8$.
5. Total Surface Area
Adding these two values together, we get the total surface area: $2 * 3.14 * (1.625)^2 + 2 * 3.14 * 1.625 * 8$.
6. Final Calculation
Calculating this gives us approximately 98.2 square inches. Therefore, the amount of aluminum required to make the soda can is approximately 98.2 square inches.
7. Final Answer
The amount of aluminum required to make the soda can is approximately $\boxed{98.2}$ square inches.
### Examples
Understanding surface area is crucial in manufacturing. For example, when designing packaging for food or beverages, knowing the surface area helps determine the amount of material needed, reducing waste and cost. This also applies to architecture, where calculating the surface area of walls and roofs helps estimate the amount of paint or roofing material required. Surface area calculations are also used in thermal engineering to estimate heat transfer from objects.
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