To determine which formulas can be used to find the surface area of a right prism, we need to consider the definitions and relationships between the different components of the surface area: the bases and the lateral area.
1. Base Area (BA): This is the area of the two bases of the prism.
2. Lateral Area (LA): This is the area of all the rectangular sides that connect the bases.
3. Perimeter of the Base (p): This is the total length around the base.
4. Height (h): This is the distance between the bases.
For a right prism, the surface area (SA) can be calculated using the formula:
[tex]\[ SA = 2 \cdot BA + LA \][/tex]
Since the lateral area [tex]\(LA\)[/tex] can also be expressed in terms of the perimeter of the base and the height as [tex]\(LA = p \cdot h\)[/tex], the formula becomes:
[tex]\[ SA = 2 \cdot BA + p \cdot h \][/tex]
Given this information, let's analyze each of the provided options:
A. [tex]\( SA = p + \angle A \)[/tex]
- This formula is incorrect because [tex]\( p \)[/tex] (perimeter of the base) plus some unspecified angle [tex]\( \angle A \)[/tex] does not represent the surface area of a right prism.
B. [tex]\( SA = BA + p \cdot h \)[/tex]
- This formula is correct. In this formula, we are combining the area of the two bases ([tex]\(2 \cdot BA\)[/tex]) multiplied by the literal area, which involves the perimeter of the base and the height ([tex]\(p \cdot h\)[/tex]).
C. [tex]\( SA = \frac{1}{2} BA + LA \)[/tex]
- This formula is incorrect. The surface area formulation involves [tex]\(2 \cdot BA\)[/tex] (accounting for both bases), not [tex]\(\frac{1}{2} BA\)[/tex], and hence does not correctly represent the equation.
D. [tex]\( SA = BA - LA \)[/tex]
- This formula is incorrect. Subtracting the lateral area from the base area does not give the surface area of a right prism.
E. [tex]\( SA = BA + \angle A \)[/tex]
- This formula is incorrect. Similar to option A, involving an unspecified angle [tex]\( \angle A \)[/tex] does not fit into the context.
Based on the above analysis, the only correct formula is:
[tex]\[ SA = BA + p \cdot h \][/tex]
So the answer to the question is option B.
