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Given a right prism where [tex]\( p \)[/tex] is the perimeter of the base, [tex]\( h \)[/tex] is the height, [tex]\( BA \)[/tex] is the area of the bases, and [tex]\( LA \)[/tex] is the lateral area, what is the surface area?

Check all that apply:
A. [tex]\( SA = 16 - \angle A \)[/tex]
B. [tex]\( SA = \frac{1}{2} 16 + \angle A \)[/tex]
C. [tex]\( SA = BA + ph \)[/tex]
D. [tex]\( SA = BA + \angle A \)[/tex]
E. [tex]\( SA = p + \angle A \)[/tex]



Answer :

GinnyAnswer
To determine which of the given options correctly describes the formula for the surface area of a right prism, let's begin by understanding the components involved and the structure of a right prism:

1. Components:
- [tex]\( p \)[/tex]: Perimeter of the base
- [tex]\( h \)[/tex]: Height of the prism
- [tex]\( B_A \)[/tex]: Area of the base
- [tex]\( I \)[/tex]: This seems to be a typo or an unnecessary element in the context of the question

2. Formula for Surface Area of a Right Prism:
- The surface area (SA) of a right prism includes the lateral area and the areas of the two bases.
- The lateral area is the perimeter of the base times the height, i.e., [tex]\( ph \)[/tex].
- We have two bases, so their total area would be [tex]\( 2 \times B_A \)[/tex].

Hence, the surface area formula would be:
[tex]\[ SA = 2 \times B_A + ph \][/tex]

With this understanding:

- Option A: [tex]\( SA = 16 - \angle A \)[/tex]
- There is no correlation between surface area and such a subtraction involving an angle measure. This is incorrect.

- Option B: [tex]\( SA = \frac{1}{2} 16 + \angle A \)[/tex]
- This option doesn't make sense in the context of calculating a surface area. There is no connection to the given variables. This is incorrect.

- Option C: [tex]\( SA = B_A + ph \)[/tex]
- This closely resembles our derived formula (excluding the factor of 2 needed for the areas of both bases). Although our formula should ideally be [tex]\( SA = 2 \times B_A + ph \)[/tex], if we only consider a single base area, it still holds a logical position. This is the closest and most relevant representation.

- Option D: [tex]\( SA = B_A + \angle A \)[/tex]
- Involving an angle measure does not fit in any way with the standard formula for the surface area of a prism. This is incorrect.

- Option E: [tex]\( SA = p + \angle A \)[/tex]
- Again, involving an angle measure and the addition of the perimeter alone is irrelevant to surface area calculation. This is incorrect.

Concluding, the option that provides the best possible formula for the surface area of the right prism, given the choices, is:

Option C: [tex]\( SA = B_A + ph \)[/tex]

Thus, the correct answer is (C).

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