To find the surface area of the hat box, we first need to determine the shape and dimensions involved.
Assuming that the hat box is a cylinder with a circular top and bottom:
- Dimensions:
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Radius (r): This seems to be sometimes misunderstood in given values. If you have a circular base and the diameter is mentioned as 30 cm, then the radius (r) would be: \[ r = \frac{diameter}{2} = \frac{30 \text{ cm}}{2} = 15 \text{ cm} \]
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Height (h): Given as 10 cm for the height of the box.
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The formula for the surface area (SA) of a cylinder is: \[ SA = 2\pi r^2 + 2\pi rh \] Where:
- \(2\pi r^2\) is the area of the two circular bases,
- \(2\pi rh\) is the area of the side (lateral surface area).
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Calculating the Areas:
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Area of the Bases: \[ 2\pi r^2 = 2 \times 3.14 \times (15 \text{ cm})^2 \] \[ = 2 \times 3.14 \times 225 \text{ cm}^2 \] \[ = 2 \times 706.5 \text{ cm}^2 = 1413 \text{ cm}^2 \]
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Lateral Surface Area: \[ 2\pi rh = 2 \times 3.14 \times 15 \text{ cm} \times 10 \text{ cm} \] \[ = 2 \times 3.14 \times 150 \text{ cm}^2 \] \[ = 2 \times 471 \text{ cm}^2 = 942 \text{ cm}^2 \]
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Total Surface Area: \[ SA = 1413 \text{ cm}^2 + 942 \text{ cm}^2 \] \[ = 2355 \text{ cm}^2 \]
This suggests a miscalculation in the parameters or dimensions may have originally been provided.
However, based on the given options, the closest one may be that of the total calculated surface area must be adjusted. The stated options like 2,512 cm² is also present. However, the valid calculation aligns around 2355 cm². None of the provided options correctly matches.
As it stands unaltered because calculation does suggest dimensions contrast potentially. It'd be prudent to validate, whether in height, width, or dimensional description along needed recalculation avenues for an accurate tally, thus prompt towards logical review on circular considerations from any net to height spacing minus coverage direct.