Question
Taylor is gift wrapping a hat box. Determine the surface area using the net of the hatbox at right.
Use 3.14 for pi (𝜋).
the top of the box is 10cm and the base is 30cm
(2 points)
Responses
2,512 cm2
2,512 cm2
628 cm2
628 cm2
1884 cm2
1884 cm2
896 cm2
Use 3.14 for pi (𝜋).
the top of the box is 10cm and the base is 30cm
(2 points)
Responses
2,512 cm2
2,512 cm2
628 cm2
628 cm2
1884 cm2
1884 cm2
896 cm2
Answers
GPT-4o mini
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:
1. 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.
2. The area of the top (which is a circle):
\[
\text{Area of the top} = \pi r^2
\]
3. The area of the base (which is also a circle):
\[
\text{Area of the base} = \pi r^2
\]
4. 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.
1. **Calculate the lateral surface area**:
\[
\text{Lateral Surface Area} = 2 \pi r h = 2 \times 3.14 \times 15 \times 10
\]
\[
= 2 \times 3.14 \times 150 = 942 \text{ cm}^2
\]
2. **Calculate the area of one circular base**:
\[
\text{Area of the base} = \pi r^2 = 3.14 \times 15^2 = 3.14 \times 225 = 706.5 \text{ cm}^2
\]
3. **Since we want both the top and base areas**:
\[
\text{Total area of top and base} = 706.5 + 706.5 = 1413 \text{ cm}^2
\]
4. **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 that’s 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.
1. 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.
2. The area of the top (which is a circle):
\[
\text{Area of the top} = \pi r^2
\]
3. The area of the base (which is also a circle):
\[
\text{Area of the base} = \pi r^2
\]
4. 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.
1. **Calculate the lateral surface area**:
\[
\text{Lateral Surface Area} = 2 \pi r h = 2 \times 3.14 \times 15 \times 10
\]
\[
= 2 \times 3.14 \times 150 = 942 \text{ cm}^2
\]
2. **Calculate the area of one circular base**:
\[
\text{Area of the base} = \pi r^2 = 3.14 \times 15^2 = 3.14 \times 225 = 706.5 \text{ cm}^2
\]
3. **Since we want both the top and base areas**:
\[
\text{Total area of top and base} = 706.5 + 706.5 = 1413 \text{ cm}^2
\]
4. **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 that’s 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.