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Which of the following formulas would find the surface area of a right cone where [tex]\(s\)[/tex] is the slant height, [tex]\(r\)[/tex] is the radius, [tex]\(LA\)[/tex] is the lateral area, and [tex]\(BA\)[/tex] is the base area?

Check all that apply.
A. [tex]\(BA + 2 \pi r^2\)[/tex]
B. [tex]\(BA + LA\)[/tex]
C. [tex]\(2 \pi r^2 + 2 \pi r h\)[/tex]
D. [tex]\(2 LA + \pi r^2\)[/tex]
E. [tex]\(\pi r^2 + \pi r s\)[/tex]

Sagot :

GinnyAnswer
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].
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