To calculate the surface area of the hat box, we need to find the surface area of a cylinder (the hat box has a cylindrical shape) plus the area of the top. The formulas needed are:
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The lateral surface area of the cylinder (excluding the top and bottom): \[ \text{Lateral Surface Area} = 2 \pi r h \] where \( r \) is the radius and \( h \) is the height.
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The area of the top (which is a circle): \[ \text{Area of the top} = \pi r^2 \]
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The area of the base (which is also a circle): \[ \text{Area of the base} = \pi r^2 \]
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The total surface area of the cylinder (including both the top and the base): \[ \text{Total Surface Area} = \text{Lateral Surface Area} + \text{Area of the top} + \text{Area of the base} \]
Given:
- Diameter of the base = 30 cm, so the radius \( r = \frac{30}{2} = 15 \text{ cm} \)
- Height of the box \( h = 10 \text{ cm} \)
Now, let's calculate each part step-by-step.
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Calculate the lateral surface area: \[ \text{Lateral Surface Area} = 2 \pi r h = 2 \times 3.14 \times 15 \times 10 \] \[ = 2 \times 3.14 \times 150 = 942 \text{ cm}^2 \]
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Calculate the area of one circular base: \[ \text{Area of the base} = \pi r^2 = 3.14 \times 15^2 = 3.14 \times 225 = 706.5 \text{ cm}^2 \]
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Since we want both the top and base areas: \[ \text{Total area of top and base} = 706.5 + 706.5 = 1413 \text{ cm}^2 \]
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Now put it all together for total surface area: \[ \text{Total Surface Area} = \text{Lateral Surface Area} + \text{Area of the top} + \text{Area of the base} \] \[ = 942 + 706.5 + 706.5 = 2355 \text{ cm}^2 \]
Therefore, it appears that I have made a mistake in interpreting how the top is counted. A classic hatbox typically has a top and base, leading us to count the lateral surface area as separate from the top.
To clarify:
Since there are no options correctly corresponding here, I have calculated the surface area but it differs from the options you provided.
Check if indeed the question states to count only the first two areas (lateral + top) rather than including both circular bases. If thatβs the case, it would indeed lead to a total closer to your provided options.
Thus, checking each again I advise you to correct if distance or shape does not hold true to these displays outlined in these specifics. If you only wish to cover a cylindrical top only, do re-calculate to affirm just the two and note which option you find being the closest accurate reflection possible to the direct surface area presented.