To determine the surface area of a hat box, which is essentially a cylinder with a top and bottom (lid), we need to calculate the lateral surface area and the area of the top and bottom.
The formula for the surface area \( A \) of a cylinder is:
\[ A = 2\pi r h + 2\pi r^2 \]
where:
- \( r \) is the radius of the base of the cylinder,
- \( h \) is the height of the cylinder.
From the problem, we have:
- Height \( h = 30 \) cm,
- Diameter of the hat box = 10 cm, so the radius \( r = \frac{10}{2} = 5 \) cm.
Now, we can calculate the surface area:
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Calculate the lateral surface area: \[ \text{Lateral Surface Area} = 2\pi rh = 2 \times 3.14 \times 5 \times 30 \] \[ = 2 \times 3.14 \times 150 = 942 \text{ cm}^2 \]
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Calculate the area of the top and bottom (2 circles): \[ \text{Area of top and bottom} = 2\pi r^2 = 2 \times 3.14 \times 5^2 \] \[ = 2 \times 3.14 \times 25 = 157 \text{ cm}^2 \]
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Total surface area: \[ A = 942 \text{ cm}^2 + 157 \text{ cm}^2 = 1099 \text{ cm}^2 \]
It appears that my calculations yield 1099 cm², but based on your multiple choice options, I may need to check the numbers against an expected answer.
Given this situation, please check your given dimensions and calculations: The original values produced a total of 1099 cm², which does not match any of the provided options. Please ensure that the values are correct or reevaluate if any additional data is needed.