To determine the correct formula for the surface area (SA) of a right prism, let's analyze each option step-by-step using the parameters given.
### Given Parameters:
- [tex]\( p \)[/tex]: Perimeter of the base
- [tex]\( h \)[/tex]: Height of the prism
- [tex]\( BA \)[/tex]: Area of the bases (each base has this area)
- [tex]\( LA \)[/tex]: Lateral area of the prism
### Surface Area (SA) Formula:
The surface area [tex]\( SA \)[/tex] of a right prism is the sum of the lateral area and the total area of the two bases. The correct formula for calculating the surface area of the prism is:
[tex]\[ SA = LA + 2 \cdot BA \][/tex]
Let's scrutinize each option provided:
### Option A: [tex]\( SA = BA + p \cdot h \)[/tex]
- In this option, the formula tries to sum the area of one base and the product of the perimeter and the height.
- [tex]\( p \cdot h \)[/tex] gives the lateral area (which is correct), but the formula is missing the area of the second base.
- Therefore, [tex]\( SA = BA + p \cdot h \)[/tex] is incorrect because it does not include the area of both bases.
### Option B: [tex]\( SA = \frac{1}{2} \cdot 16 + LA \)[/tex]
- This option does not logically make sense for a general formula for the surface area of a right prism.
- The term [tex]\(\frac{1}{2} \cdot 16\)[/tex] seems arbitrary and does not align with the correct surface area formula structure unless specific numerical values are provided that fit exactly within a specific context.
- This option is generally incorrect for a universal formula.
### Option C: [tex]\( SA = p + \angle A \)[/tex]
- This option is incorrect because [tex]\( \angle A \)[/tex] (an angle) is irrelevant to the calculation of surface area.
- Adding the perimeter to an angle does not conform to any known formula for surface area.
### Option D: [tex]\( SA = 16 - LA \)[/tex]
- This option implies that the surface area is derived by subtracting the lateral area from 16, which is arbitrary and does not fit with the known formula for surface area.
- This is incorrect because it does not correlate with the formula which sums areas rather than subtracting.
### Summary:
- None of the given options (A, B, C, or D) are correct.
- The correct formula for the surface area of a right prism is:
[tex]\[ SA = LA + 2 \cdot BA \][/tex]
Given the analysis of the options provided, we conclude that none of the options offered match the correct formula. Hence, the answer is that none of the provided options are correct.
