Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Discover a wealth of knowledge from experts across different disciplines on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.