Answer :
To match the given surface areas of the shoeboxes to their respective dimensions, let's calculate the surface area for each box first. The formula for the surface area of a rectangular prism is as follows:
[tex]\[ \text{Surface Area} = 2(l \times w + l \times h + w \times h) \][/tex]
Where \( l \) is the length, \( w \) is the width, and \( h \) is the height of the box.
### Calculations:
1. Dimensions: 10 in × 6 in × 6 in
[tex]\[ \text{Surface Area} = 2(10 \times 6 + 10 \times 6 + 6 \times 6) \][/tex]
[tex]\[ \text{Surface Area} = 2(60 + 60 + 36) \][/tex]
[tex]\[ \text{Surface Area} = 2 \times 156 \][/tex]
[tex]\[ \text{Surface Area} = 312 \, \text{in}^2\][/tex]
2. Dimensions: 14 in × 5 in × 5 in
[tex]\[ \text{Surface Area} = 2(14 \times 5 + 14 \times 5 + 5 \times 5) \][/tex]
[tex]\[ \text{Surface Area} = 2(70 + 70 + 25) \][/tex]
[tex]\[ \text{Surface Area} = 2 \times 165 \][/tex]
[tex]\[ \text{Surface Area} = 330 \, \text{in}^2\][/tex]
3. Dimensions: 16 in × 6 in × 4 in
[tex]\[ \text{Surface Area} = 2(16 \times 6 + 16 \times 4 + 6 \times 4) \][/tex]
[tex]\[ \text{Surface Area} = 2(96 + 64 + 24) \][/tex]
[tex]\[ \text{Surface Area} = 2 \times 184 \][/tex]
[tex]\[ \text{Surface Area} = 368 \, \text{in}^2\][/tex]
4. Dimensions: 12 in × 5 in × 6 in
[tex]\[ \text{Surface Area} = 2(12 \times 5 + 12 \times 6 + 5 \times 6) \][/tex]
[tex]\[ \text{Surface Area} = 2(60 + 72 + 30) \][/tex]
[tex]\[ \text{Surface Area} = 2 \times 162 \][/tex]
[tex]\[ \text{Surface Area} = 324 \, \text{in}^2\][/tex]
### Matching Results:
- The box with dimensions 10 in × 6 in × 6 in has a surface area of 312 in².
- The box with dimensions 14 in × 5 in × 5 in has a surface area of 330 in².
- The box with dimensions 16 in × 6 in × 4 in has a surface area of 368 in².
- The box with dimensions 12 in × 5 in × 6 in has a surface area of 324 in².
So, the matches are:
- \( 312 \, \text{in}^2 \) matches with \( 10 \, \text{in} \times 6 \, \text{in} \times 6 \, \text{in} \)
- \( 330 \, \text{in}^2 \) matches with \( 14 \, \text{in} \times 5 \, \text{in} \times 5 \, \text{in} \)
- \( 368 \, \text{in}^2 \) matches with \( 16 \, \text{in} \times 6 \, \text{in} \times 4 \, \text{in} \)
- [tex]\( 324 \, \text{in}^2 \)[/tex] matches with [tex]\( 12 \, \text{in} \times 5 \, \text{in} \times 6 \, \text{in} \)[/tex]
[tex]\[ \text{Surface Area} = 2(l \times w + l \times h + w \times h) \][/tex]
Where \( l \) is the length, \( w \) is the width, and \( h \) is the height of the box.
### Calculations:
1. Dimensions: 10 in × 6 in × 6 in
[tex]\[ \text{Surface Area} = 2(10 \times 6 + 10 \times 6 + 6 \times 6) \][/tex]
[tex]\[ \text{Surface Area} = 2(60 + 60 + 36) \][/tex]
[tex]\[ \text{Surface Area} = 2 \times 156 \][/tex]
[tex]\[ \text{Surface Area} = 312 \, \text{in}^2\][/tex]
2. Dimensions: 14 in × 5 in × 5 in
[tex]\[ \text{Surface Area} = 2(14 \times 5 + 14 \times 5 + 5 \times 5) \][/tex]
[tex]\[ \text{Surface Area} = 2(70 + 70 + 25) \][/tex]
[tex]\[ \text{Surface Area} = 2 \times 165 \][/tex]
[tex]\[ \text{Surface Area} = 330 \, \text{in}^2\][/tex]
3. Dimensions: 16 in × 6 in × 4 in
[tex]\[ \text{Surface Area} = 2(16 \times 6 + 16 \times 4 + 6 \times 4) \][/tex]
[tex]\[ \text{Surface Area} = 2(96 + 64 + 24) \][/tex]
[tex]\[ \text{Surface Area} = 2 \times 184 \][/tex]
[tex]\[ \text{Surface Area} = 368 \, \text{in}^2\][/tex]
4. Dimensions: 12 in × 5 in × 6 in
[tex]\[ \text{Surface Area} = 2(12 \times 5 + 12 \times 6 + 5 \times 6) \][/tex]
[tex]\[ \text{Surface Area} = 2(60 + 72 + 30) \][/tex]
[tex]\[ \text{Surface Area} = 2 \times 162 \][/tex]
[tex]\[ \text{Surface Area} = 324 \, \text{in}^2\][/tex]
### Matching Results:
- The box with dimensions 10 in × 6 in × 6 in has a surface area of 312 in².
- The box with dimensions 14 in × 5 in × 5 in has a surface area of 330 in².
- The box with dimensions 16 in × 6 in × 4 in has a surface area of 368 in².
- The box with dimensions 12 in × 5 in × 6 in has a surface area of 324 in².
So, the matches are:
- \( 312 \, \text{in}^2 \) matches with \( 10 \, \text{in} \times 6 \, \text{in} \times 6 \, \text{in} \)
- \( 330 \, \text{in}^2 \) matches with \( 14 \, \text{in} \times 5 \, \text{in} \times 5 \, \text{in} \)
- \( 368 \, \text{in}^2 \) matches with \( 16 \, \text{in} \times 6 \, \text{in} \times 4 \, \text{in} \)
- [tex]\( 324 \, \text{in}^2 \)[/tex] matches with [tex]\( 12 \, \text{in} \times 5 \, \text{in} \times 6 \, \text{in} \)[/tex]
