To find the surface area [tex]\( S A \)[/tex] of a right prism, we need to consider both the areas of the bases and the lateral faces. The given quantities are:
- [tex]\( p \)[/tex]: the perimeter of the base
- [tex]\( h \)[/tex]: the height of the prism
- [tex]\( B A \)[/tex]: the area of one base
- [tex]\( L A \)[/tex]: the lateral area
### Surface Area of a Right Prism
The surface area of a right prism [tex]\( S A \)[/tex] is calculated by summing the areas of all the faces. Specifically, it can be calculated as follows:
1. Bases Area Contribution: The prism has two bases, so the total area of the bases is [tex]\( 2 \times B A \)[/tex].
2. Lateral Area Contribution: The lateral area [tex]\( L A \)[/tex] encompasses all the side faces of the prism.
So, the formula for the surface area of the prism becomes:
[tex]\[ S A = 2 \times B A + L A \][/tex]
### Checking Each Given Option:
A. [tex]\( S A = B A + L A \)[/tex]
- This formula only considers one base area and the lateral area. This is not correct because it misses the area of the second base.
- Not correct.
B. [tex]\( S A = \frac{1}{2} 16 + L A \)[/tex]
- This does not align with the concepts for areas in the context of a right prism. [tex]\( \frac{1}{2} 16 \)[/tex] seems arbitrary and irrelevant here.
- Not correct.
C. [tex]\( S A = 16 - \angle A \)[/tex]
- This formula mixes an area operation with an angle, which does not make sense for calculating surface area.
- Not correct.
D. [tex]\( S A = B A + p h \)[/tex]
- This formula combines the base area [tex]\( B A \)[/tex] with the product of the perimeter of the base [tex]\( p \)[/tex] and the height [tex]\( h \)[/tex].
- The term [tex]\( p h \)[/tex] represents the lateral area for a right prism (if [tex]\( p \)[/tex] denotes the total perimeter and [tex]\( h \)[/tex] the height).
- This simplifies to [tex]\( S A = 2 \times B A + L A \)[/tex], which matches the correct formula.
- Correct.
E. [tex]\( S A = p + L A \)[/tex]
- This formula adds the perimeter [tex]\( p \)[/tex] directly to the lateral area, which does not match the traditional calculation of surface area for a right prism.
- Not correct.
### Correct Options
From the analysis above, the correct answer is:
- [tex]\( D. S A = B A + p h \)[/tex]
Thus, the proper formula(s) for the surface area provided by the options is D. Only this captures the total surface area correctly for a right prism.
