To determine which formulas correctly represent the surface area of a right cone, we should recall the formula for the surface area of a right cone.
The surface area [tex]\(A\)[/tex] of a right cone is given by the sum of the base area plus the lateral area. The base of a right cone is a circle with area [tex]\( \pi r^2 \)[/tex], and the lateral surface area is [tex]\( \pi r s \)[/tex] where [tex]\( r \)[/tex] is the radius and [tex]\( s \)[/tex] is the slant height of the cone. Therefore, the surface area [tex]\(A\)[/tex] is:
[tex]\[ A = \pi r^2 + \pi r s \][/tex]
Now, let's analyze the given formulas to check which ones match the expression for the surface area of a right cone:
A. [tex]\( B A + 2 \pi r^2 \)[/tex]
- This formula includes the base area ([tex]\( B A \)[/tex]) plus an additional area of [tex]\( 2 \pi r^2 \)[/tex]. This does not match the required expression since it adds the base area twice incorrectly.
B. [tex]\( B A + L A \)[/tex]
- Here, [tex]\( B A \)[/tex] represents the base area, and [tex]\( L A \)[/tex] represents the lateral area. Summing these gives us the total surface area of the right cone; hence, this formula is correct.
C. [tex]\( 2 \pi r^2 + 2 \pi r h \)[/tex]
- This formula includes twice the base area and a term involving the radius and height. It does not match the required formula, as it misrepresents the components of the surface area of a right cone.
D. [tex]\( 2 L A + \pi I^2 \)[/tex]
- This formula incorrectly doubles the lateral area and adds the square of [tex]\( I \)[/tex] (which is undefined in this context). Thus, it does not represent the surface area of a cone.
E. [tex]\( \pi r^2 + \pi r s \)[/tex]
- This formula exactly matches the standard expression for the surface area of a right cone: the sum of the base area and the lateral area. Therefore, this formula is correct.
In conclusion, the formulas that find the surface area of a right cone are:
- [tex]\( \textbf{B:} \, B A + L A \)[/tex]
- [tex]\( \textbf{E:} \, \pi r^2 + \pi r s \)[/tex]
So, the correct answers are [tex]\( \textbf{B} \)[/tex] and [tex]\( \textbf{E} \)[/tex].
