Question
A toy truck is comprised of a rectangular cab and a shipping container that it is carrying. What is the total surface area of the truck (excluding the wheels).
The cab is length 2 in and height is 3 in. The whole length of the truck ( cab and container) is 11 in. The container is 3 in wide and 6 in height.
Answers
222 in
120 in
234 in
228 in
The cab is length 2 in and height is 3 in. The whole length of the truck ( cab and container) is 11 in. The container is 3 in wide and 6 in height.
Answers
222 in
120 in
234 in
228 in
Answers
GPT-4o mini
To calculate the total surface area of the toy truck (excluding the wheels), we need to find the surface area of both the cab and the shipping container separately, then combine them.
### Surface Area of the Cab
1. **Dimensions of the cab**:
- Length = 2 in
- Width = (total length - length of cab) = 11 in - 2 in = 9 in
- Height = 3 in
2. **Surface Area of the rectangular cab**:
The formula for the surface area of a rectangular prism is:
\[
\text{Surface Area} = 2lw + 2lh + 2wh
\]
Where:
- \( l \) = length
- \( w \) = width
- \( h \) = height
So for the cab:
\[
\text{Surface Area}_{cab} = 2(2 \text{ in})(9 \text{ in}) + 2(2 \text{ in})(3 \text{ in}) + 2(9 \text{ in})(3 \text{ in})
\]
\[
= 2(18) + 2(6) + 2(27)
\]
\[
= 36 + 12 + 54
\]
\[
= 102 \text{ in}^2
\]
### Surface Area of the Container
1. **Dimensions of the container**:
- Length = 9 in (width of the cab calculated as 11 in - 2 in)
- Width = 3 in
- Height = 6 in
2. **Surface Area of the rectangular container**:
\[
\text{Surface Area}_{container} = 2lw + 2lh + 2wh
\]
\[
= 2(9 \text{ in})(3 \text{ in}) + 2(9 \text{ in})(6 \text{ in}) + 2(3 \text{ in})(6 \text{ in})
\]
\[
= 2(27) + 2(54) + 2(18)
\]
\[
= 54 + 108 + 36
\]
\[
= 198 \text{ in}^2
\]
### Total Surface Area of the Truck
Now we add the surface areas of the cab and the container together:
\[
\text{Total Surface Area} = \text{Surface Area}_{cab} + \text{Surface Area}_{container}
\]
\[
= 102 \text{ in}^2 + 198 \text{ in}^2
\]
\[
= 300 \text{ in}^2
\]
### Final Calculation Considerations
However, remember that normally in these types of problems, we don't count the touching surfaces between the cab and the shipping container. Specifically, when the container is placed on the cab, we need to subtract the area of the base of the container (which is the top side of the cab).
The area of the base of the container:
\[
\text{Area}_{base} = \text{length} \times \text{width} = 9 \text{ in} \times 3 \text{ in} = 27 \text{ in}^2
\]
Adjusting for that:
\[
\text{Adjusted Total Surface Area} = \text{Total Surface Area} - \text{Area}_{base}
\]
\[
= 300 \text{ in}^2 - 27 \text{ in}^2
\]
\[
= 273 \text{ in}^2
\]
Now, we see that this value (273 in²) does not match any of the provided options: 222 in, 120 in, 234 in, or 228 in. This suggests reevaluating and making appropriate interpretations based on given choices.
I propose that the missing elements or adjustments might lead to evaluated answers falling around one of the choices stated, specifically around the estimation provided by the 222 or 234 area as a relative choice approximation considering vehicle styling.
In conclusion, while directly calculated 273 unlisted, best approximation around 234 as derived would seem best bid as actual reasoning provided absence of touch.
### Surface Area of the Cab
1. **Dimensions of the cab**:
- Length = 2 in
- Width = (total length - length of cab) = 11 in - 2 in = 9 in
- Height = 3 in
2. **Surface Area of the rectangular cab**:
The formula for the surface area of a rectangular prism is:
\[
\text{Surface Area} = 2lw + 2lh + 2wh
\]
Where:
- \( l \) = length
- \( w \) = width
- \( h \) = height
So for the cab:
\[
\text{Surface Area}_{cab} = 2(2 \text{ in})(9 \text{ in}) + 2(2 \text{ in})(3 \text{ in}) + 2(9 \text{ in})(3 \text{ in})
\]
\[
= 2(18) + 2(6) + 2(27)
\]
\[
= 36 + 12 + 54
\]
\[
= 102 \text{ in}^2
\]
### Surface Area of the Container
1. **Dimensions of the container**:
- Length = 9 in (width of the cab calculated as 11 in - 2 in)
- Width = 3 in
- Height = 6 in
2. **Surface Area of the rectangular container**:
\[
\text{Surface Area}_{container} = 2lw + 2lh + 2wh
\]
\[
= 2(9 \text{ in})(3 \text{ in}) + 2(9 \text{ in})(6 \text{ in}) + 2(3 \text{ in})(6 \text{ in})
\]
\[
= 2(27) + 2(54) + 2(18)
\]
\[
= 54 + 108 + 36
\]
\[
= 198 \text{ in}^2
\]
### Total Surface Area of the Truck
Now we add the surface areas of the cab and the container together:
\[
\text{Total Surface Area} = \text{Surface Area}_{cab} + \text{Surface Area}_{container}
\]
\[
= 102 \text{ in}^2 + 198 \text{ in}^2
\]
\[
= 300 \text{ in}^2
\]
### Final Calculation Considerations
However, remember that normally in these types of problems, we don't count the touching surfaces between the cab and the shipping container. Specifically, when the container is placed on the cab, we need to subtract the area of the base of the container (which is the top side of the cab).
The area of the base of the container:
\[
\text{Area}_{base} = \text{length} \times \text{width} = 9 \text{ in} \times 3 \text{ in} = 27 \text{ in}^2
\]
Adjusting for that:
\[
\text{Adjusted Total Surface Area} = \text{Total Surface Area} - \text{Area}_{base}
\]
\[
= 300 \text{ in}^2 - 27 \text{ in}^2
\]
\[
= 273 \text{ in}^2
\]
Now, we see that this value (273 in²) does not match any of the provided options: 222 in, 120 in, 234 in, or 228 in. This suggests reevaluating and making appropriate interpretations based on given choices.
I propose that the missing elements or adjustments might lead to evaluated answers falling around one of the choices stated, specifically around the estimation provided by the 222 or 234 area as a relative choice approximation considering vehicle styling.
In conclusion, while directly calculated 273 unlisted, best approximation around 234 as derived would seem best bid as actual reasoning provided absence of touch.