To calculate the surface area of the hat box, we need to consider it as a cylinder with a top and a base. The formula for the surface area \( SA \) of a cylinder is given by:
\[ SA = 2\pi r^2 + 2\pi rh \]
Where:
- \( r \) is the radius
- \( h \) is the height
Given the dimensions:
- The diameter of the base is 30 cm, therefore, the radius \( r = \frac{30}{2} = 15 \) cm.
- The height of the box (which is also the height of the cylinder) is 10 cm.
Now we can calculate the surface area:
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Calculate the area of the top and base: \[ 2\pi r^2 = 2 \times 3.14 \times (15)^2 = 2 \times 3.14 \times 225 = 1413 , \text{cm}^2 \]
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Calculate the lateral surface area: \[ 2\pi rh = 2 \times 3.14 \times 15 \times 10 = 2 \times 3.14 \times 150 = 942 , \text{cm}^2 \]
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Add both areas together to get the total surface area: \[ SA = 1413 + 942 = 2355 , \text{cm}^2 \]
It seems that the calculation might include a mistake as none of the answers correspond to \( 2355 , \text{cm}^2 \). Let's double-check the calculations:
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Top and base area calculation: \[ 2\pi r^2 = 2 \times 3.14 \times 15^2 = 2 \times 3.14 \times 225 = 1413 , \text{cm}^2 \]
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Lateral surface area: \[ 2\pi rh = 2 \times 3.14 \times 15 \times 10 = 942 , \text{cm}^2 \]
Adding them gives: \[ 1413 + 942 = 2355 , \text{cm}^2 \]
Since our calculation still does not match any of the options, please verify that the dimensions (diameter of the base and height) are correct. Should you need the dimensions to be adjusted or you need a different approach, let me know! However, based on the calculations, none of the provided options match.