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Sagot :
Sure, let's analyze the given expression step by step to identify the errors Devon made.
The expression Devon used to find the surface area is:
[tex]\[ 2\left[(9 \times 20) + (9 \times 34)\right] + (34 \times 20) + 2 \left[\frac{1}{2} (20 \times 24)\right] + 2 \left[\frac{1}{2} (34 \times 26)\right] \][/tex]
### Step-by-Step Analysis:
1. First Term: [tex]\(2(9 \times 20)\)[/tex]
- This term calculates the area of two rectangles with dimensions [tex]\(9 \times 20\)[/tex].
2. Second Term: [tex]\(2(9 \times 34)\)[/tex]
- This term calculates the area of two rectangles with dimensions [tex]\(9 \times 34\)[/tex].
3. Third Term: [tex]\(34 \times 20\)[/tex]
- This term represents the area of a base with dimensions [tex]\(34 \times 20\)[/tex].
4. Fourth Term: [tex]\(2 \left[\frac{1}{2} (20 \times 24)\right] \)[/tex]
- This calculates the area of two triangles with base [tex]\(20\)[/tex] and height (slant height) [tex]\(24\)[/tex].
5. Fifth Term: [tex]\(2 \left[\frac{1}{2} (34 \times 26)\right] \)[/tex]
- This calculates the area of two triangles with base [tex]\(34\)[/tex] and height (slant height) [tex]\(26\)[/tex].
### Errors Identified:
1. Incorrect Slant Height:
- When calculating the area of the triangles in the fifth term [tex]\(2 \left[\frac{1}{2} (34 \times 26)\right] \)[/tex], Devon used 26 as the slant height instead of the correct slant height of 24.
- This is an error because the slant height used in the two sets of triangles should be consistent.
2. Included the Base Rectangle:
- The term [tex]\(34 \times 20\)[/tex] represents the area of a rectangle which is the base of the figure.
- Devon included this base in his calculation, which should not be included when calculating the surface area of the lateral faces and the upper solid.
3. Omitted the Joining Rectangle:
- There is a rectangle where the upper and lower solids join together that Devon omitted in his expression.
- This omission means that there was a missing component in the surface area calculation.
### Conclusion:
Based on our detailed analysis above, Devon made the following errors:
1. He used 26 instead of 24 as the slant height of one set of lateral faces.
2. He included the [tex]\(34 \times 20\)[/tex] base of the figure.
3. He omitted the rectangle where the upper and lower solids join together.
Thus, the errors Devon made are accurately represented by the following statements:
- He used 26 instead of 24 as the slant height of one set of lateral faces.
- He omitted the rectangle where the upper and lower solids join together.
- He included the [tex]\(34 \times 20\)[/tex] base of the figure.
The expression Devon used to find the surface area is:
[tex]\[ 2\left[(9 \times 20) + (9 \times 34)\right] + (34 \times 20) + 2 \left[\frac{1}{2} (20 \times 24)\right] + 2 \left[\frac{1}{2} (34 \times 26)\right] \][/tex]
### Step-by-Step Analysis:
1. First Term: [tex]\(2(9 \times 20)\)[/tex]
- This term calculates the area of two rectangles with dimensions [tex]\(9 \times 20\)[/tex].
2. Second Term: [tex]\(2(9 \times 34)\)[/tex]
- This term calculates the area of two rectangles with dimensions [tex]\(9 \times 34\)[/tex].
3. Third Term: [tex]\(34 \times 20\)[/tex]
- This term represents the area of a base with dimensions [tex]\(34 \times 20\)[/tex].
4. Fourth Term: [tex]\(2 \left[\frac{1}{2} (20 \times 24)\right] \)[/tex]
- This calculates the area of two triangles with base [tex]\(20\)[/tex] and height (slant height) [tex]\(24\)[/tex].
5. Fifth Term: [tex]\(2 \left[\frac{1}{2} (34 \times 26)\right] \)[/tex]
- This calculates the area of two triangles with base [tex]\(34\)[/tex] and height (slant height) [tex]\(26\)[/tex].
### Errors Identified:
1. Incorrect Slant Height:
- When calculating the area of the triangles in the fifth term [tex]\(2 \left[\frac{1}{2} (34 \times 26)\right] \)[/tex], Devon used 26 as the slant height instead of the correct slant height of 24.
- This is an error because the slant height used in the two sets of triangles should be consistent.
2. Included the Base Rectangle:
- The term [tex]\(34 \times 20\)[/tex] represents the area of a rectangle which is the base of the figure.
- Devon included this base in his calculation, which should not be included when calculating the surface area of the lateral faces and the upper solid.
3. Omitted the Joining Rectangle:
- There is a rectangle where the upper and lower solids join together that Devon omitted in his expression.
- This omission means that there was a missing component in the surface area calculation.
### Conclusion:
Based on our detailed analysis above, Devon made the following errors:
1. He used 26 instead of 24 as the slant height of one set of lateral faces.
2. He included the [tex]\(34 \times 20\)[/tex] base of the figure.
3. He omitted the rectangle where the upper and lower solids join together.
Thus, the errors Devon made are accurately represented by the following statements:
- He used 26 instead of 24 as the slant height of one set of lateral faces.
- He omitted the rectangle where the upper and lower solids join together.
- He included the [tex]\(34 \times 20\)[/tex] base of the figure.
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