To find valid formulas for the surface area (SA) of a regular pyramid, we must recall the definitions and standard relationships for the quantities involved. Let's analyze each given expression:
1. A. [tex]\( SA = BA + 2LA \)[/tex]
- The surface area of a pyramid consists of the base area [tex]\( BA \)[/tex] and the lateral area [tex]\( LA \)[/tex]. The [tex]\( 2LA \)[/tex] term implies that the lateral area is being doubled, which is not correct for the surface area calculation. Hence, this is incorrect.
2. B. [tex]\( SA = BA + \frac{1}{2}ps \)[/tex]
- Here, [tex]\( p \)[/tex] is the perimeter of the base, and [tex]\( s \)[/tex] is the slant height. The term [tex]\( \frac{1}{2}ps \)[/tex] correctly represents the lateral area of a regular pyramid for such a calculation. Therefore, this expression is valid.
3. C. [tex]\( SA = BA + LA \)[/tex]
- This formula correctly calculates the surface area by summing the base area [tex]\( BA \)[/tex] and the lateral area [tex]\( LA \)[/tex]. Hence, this is a valid expression.
4. D. [tex]\( SA = 8 \cdot LA \)[/tex]
- This implies that the surface area is eight times the lateral area. There is no standard relationship supporting this formula for pyramids. Hence, this is incorrect.
5. E. [tex]\( SA = 2BA + \frac{1}{2}ps \)[/tex]
- This expression incorrectly doubles the base area when calculating the surface area of the pyramid, which does not align with standard formulas. Therefore, this is incorrect.
Thus, the valid formulas for the surface area of a regular pyramid are:
B. [tex]\( SA = BA + \frac{1}{2}ps \)[/tex]
C. [tex]\( SA = BA + LA \)[/tex]
The solution hence checks formulas B and C as valid, leading to the result:
[tex]\[ \boxed{[2, 3]} \][/tex]
