Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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]
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
