Answer :
To determine the correct formulas for the surface area [tex]\( SA \)[/tex] of a right prism involving the variables [tex]\( p \)[/tex] (perimeter of the base), [tex]\( h \)[/tex] (height), [tex]\( BA \)[/tex] (area of the bases), and [tex]\( LA \)[/tex] (lateral area), we need to assess the given options and validate which expressions actually equate to the surface area.
Let's consider each option:
### Option A: [tex]\( SA = \frac{1}{2} \times 16 + LA \)[/tex]
This expression calculates the surface area by taking half of 16 and adding it to the lateral area:
[tex]\[ SA = \frac{1}{2} \times 16 + LA \][/tex]
Evaluating this:
[tex]\[ SA = 8 + LA \][/tex]
Given [tex]\( LA = 50 \)[/tex]:
[tex]\[ SA = 8 + 50 = 58 \][/tex]
This results in 58, which matches the numerical result provided earlier, so this is a valid expression for surface area.
### Option B: [tex]\( SA = BA + p \times h \)[/tex]
This expression calculates the surface area by adding the base area to the product of the perimeter and height:
[tex]\[ SA = BA + p \times h \][/tex]
Given [tex]\( BA = 20 \)[/tex], [tex]\( p = 10 \)[/tex], and [tex]\( h = 5 \)[/tex]:
[tex]\[ SA = 20 + 10 \times 5 = 20 + 50 = 70 \][/tex]
This results in 70, which aligns with our provided numerical result, so this is also a valid expression for surface area.
### Option C: [tex]\( SA = BA + \angle A \)[/tex]
This expression mentions adding the base area to angle [tex]\( A \)[/tex]. Surface area calculations typically don't involve angles directly unless in specialized contexts not specified here.
[tex]\[ SA = BA + \angle A \][/tex]
This equation does not provide a meaningful or standard result for surface area and is incorrect.
### Option D: [tex]\( SA = 16 - LA \)[/tex]
This expression calculates the surface area by subtracting the lateral area from 16:
[tex]\[ SA = 16 - LA \][/tex]
Given [tex]\( LA = 50 \)[/tex]:
[tex]\[ SA = 16 - 50 = -34 \][/tex]
This results in -34, which is clearly incorrect as surface area cannot be negative. Therefore, this equation is not valid for surface area.
### Option E: [tex]\( SA = p + \angle A \)[/tex]
This expression involves adding the perimeter of the base to angle [tex]\( A \)[/tex], which does not make sense in calculating surface area:
[tex]\[ SA = p + \angle A \][/tex]
Since perimeter and angles do not combine in any standard surface area formula, this is also incorrect.
### Conclusion:
The correct expressions for the surface area [tex]\( SA \)[/tex] that apply are:
- Option A: [tex]\( SA = \frac{1}{2} \times 16 + LA \)[/tex]
- Option B: [tex]\( SA = BA + p \times h \)[/tex]
Let's consider each option:
### Option A: [tex]\( SA = \frac{1}{2} \times 16 + LA \)[/tex]
This expression calculates the surface area by taking half of 16 and adding it to the lateral area:
[tex]\[ SA = \frac{1}{2} \times 16 + LA \][/tex]
Evaluating this:
[tex]\[ SA = 8 + LA \][/tex]
Given [tex]\( LA = 50 \)[/tex]:
[tex]\[ SA = 8 + 50 = 58 \][/tex]
This results in 58, which matches the numerical result provided earlier, so this is a valid expression for surface area.
### Option B: [tex]\( SA = BA + p \times h \)[/tex]
This expression calculates the surface area by adding the base area to the product of the perimeter and height:
[tex]\[ SA = BA + p \times h \][/tex]
Given [tex]\( BA = 20 \)[/tex], [tex]\( p = 10 \)[/tex], and [tex]\( h = 5 \)[/tex]:
[tex]\[ SA = 20 + 10 \times 5 = 20 + 50 = 70 \][/tex]
This results in 70, which aligns with our provided numerical result, so this is also a valid expression for surface area.
### Option C: [tex]\( SA = BA + \angle A \)[/tex]
This expression mentions adding the base area to angle [tex]\( A \)[/tex]. Surface area calculations typically don't involve angles directly unless in specialized contexts not specified here.
[tex]\[ SA = BA + \angle A \][/tex]
This equation does not provide a meaningful or standard result for surface area and is incorrect.
### Option D: [tex]\( SA = 16 - LA \)[/tex]
This expression calculates the surface area by subtracting the lateral area from 16:
[tex]\[ SA = 16 - LA \][/tex]
Given [tex]\( LA = 50 \)[/tex]:
[tex]\[ SA = 16 - 50 = -34 \][/tex]
This results in -34, which is clearly incorrect as surface area cannot be negative. Therefore, this equation is not valid for surface area.
### Option E: [tex]\( SA = p + \angle A \)[/tex]
This expression involves adding the perimeter of the base to angle [tex]\( A \)[/tex], which does not make sense in calculating surface area:
[tex]\[ SA = p + \angle A \][/tex]
Since perimeter and angles do not combine in any standard surface area formula, this is also incorrect.
### Conclusion:
The correct expressions for the surface area [tex]\( SA \)[/tex] that apply are:
- Option A: [tex]\( SA = \frac{1}{2} \times 16 + LA \)[/tex]
- Option B: [tex]\( SA = BA + p \times h \)[/tex]
