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To determine the correct formulas for the surface area (SA) of a right prism, we need to analyze each option based on the given parameters: [tex]\( p \)[/tex] (perimeter of the base), [tex]\( h \)[/tex] (height), [tex]\( BA \)[/tex] (area of the bases), and [tex]\( LA \)[/tex] (lateral area).
1. Option A: [tex]\( SA = \frac{1}{2} \cdot 16 + LA \)[/tex]
- If we examine the structure of this equation:
- The term [tex]\(\frac{1}{2} \cdot 16\)[/tex] simplifies to 8.
- Therefore, this equation simplifies to [tex]\( SA = 8 + LA \)[/tex].
- Given the logical structure, this equation correctly represents the sum of a constant and the lateral area.
- The resulting surface area in this case is determined to be [tex]\( 8 + LA \)[/tex], which aligns with the correct surface area calculation strategy for some specific cases.
2. Option B: [tex]\( SA = 16 - \angle A \)[/tex]
- This equation does not directly incorporate any of the given parameters: [tex]\( p \)[/tex], [tex]\( h \)[/tex], [tex]\( BA \)[/tex], or [tex]\( LA \)[/tex].
- It subtracts an angle [tex]\( \angle A \)[/tex] from a constant, which does not relate to the concept of surface area in the context of a prism.
3. Option C: [tex]\( SA = p + \angle A \)[/tex]
- This option adds the perimeter of the base [tex]\( p \)[/tex] to some angle [tex]\( \angle A \)[/tex].
- Neither [tex]\( p \)[/tex] nor an arbitrary angle [tex]\( \angle A \)[/tex] will give a complete or meaningful calculation of surface area of a right prism.
- This formula does not directly utilize essential components like the lateral or base areas.
4. Option D: [tex]\( SA = BA + ph \)[/tex]
- To analyze this formula:
- [tex]\( ph \)[/tex] calculates the lateral area of the prism by multiplying the perimeter of the base with the height.
- [tex]\( BA \)[/tex] represents the area of one base of the prism.
- Summing [tex]\( BA \)[/tex] and [tex]\( ph \)[/tex] corresponds to the lateral area plus the area of the bases.
- Therefore, this represents a correct calculation for the surface area of the prism.
5. Option E: [tex]\( SA = BA + LA \)[/tex]
- Breaking it down:
- [tex]\( BA \)[/tex] represents the base area.
- [tex]\( LA \)[/tex] represents the lateral area.
- Adding these two gives the surface area, since the total surface area of a right prism is the sum of the area of the bases and the lateral area.
- Hence, this formula is a valid and correct representation of the total surface area.
Conclusion:
The correct options that reflect the valid formulas for the surface area [tex]\( SA \)[/tex] are:
- Option A (simplifies to [tex]\( SA = 8 + LA \)[/tex])
- Option D ( [tex]\( SA = BA + ph \)[/tex])
- Option E ( [tex]\( SA = BA + LA \)[/tex])
Thus, the surface area calculation in the context of a right prism aligns with options A, D, and E.
1. Option A: [tex]\( SA = \frac{1}{2} \cdot 16 + LA \)[/tex]
- If we examine the structure of this equation:
- The term [tex]\(\frac{1}{2} \cdot 16\)[/tex] simplifies to 8.
- Therefore, this equation simplifies to [tex]\( SA = 8 + LA \)[/tex].
- Given the logical structure, this equation correctly represents the sum of a constant and the lateral area.
- The resulting surface area in this case is determined to be [tex]\( 8 + LA \)[/tex], which aligns with the correct surface area calculation strategy for some specific cases.
2. Option B: [tex]\( SA = 16 - \angle A \)[/tex]
- This equation does not directly incorporate any of the given parameters: [tex]\( p \)[/tex], [tex]\( h \)[/tex], [tex]\( BA \)[/tex], or [tex]\( LA \)[/tex].
- It subtracts an angle [tex]\( \angle A \)[/tex] from a constant, which does not relate to the concept of surface area in the context of a prism.
3. Option C: [tex]\( SA = p + \angle A \)[/tex]
- This option adds the perimeter of the base [tex]\( p \)[/tex] to some angle [tex]\( \angle A \)[/tex].
- Neither [tex]\( p \)[/tex] nor an arbitrary angle [tex]\( \angle A \)[/tex] will give a complete or meaningful calculation of surface area of a right prism.
- This formula does not directly utilize essential components like the lateral or base areas.
4. Option D: [tex]\( SA = BA + ph \)[/tex]
- To analyze this formula:
- [tex]\( ph \)[/tex] calculates the lateral area of the prism by multiplying the perimeter of the base with the height.
- [tex]\( BA \)[/tex] represents the area of one base of the prism.
- Summing [tex]\( BA \)[/tex] and [tex]\( ph \)[/tex] corresponds to the lateral area plus the area of the bases.
- Therefore, this represents a correct calculation for the surface area of the prism.
5. Option E: [tex]\( SA = BA + LA \)[/tex]
- Breaking it down:
- [tex]\( BA \)[/tex] represents the base area.
- [tex]\( LA \)[/tex] represents the lateral area.
- Adding these two gives the surface area, since the total surface area of a right prism is the sum of the area of the bases and the lateral area.
- Hence, this formula is a valid and correct representation of the total surface area.
Conclusion:
The correct options that reflect the valid formulas for the surface area [tex]\( SA \)[/tex] are:
- Option A (simplifies to [tex]\( SA = 8 + LA \)[/tex])
- Option D ( [tex]\( SA = BA + ph \)[/tex])
- Option E ( [tex]\( SA = BA + LA \)[/tex])
Thus, the surface area calculation in the context of a right prism aligns with options A, D, and E.
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