Answer :
To determine the surface area of a rectangular prism with dimensions [tex]\( x \)[/tex] units, [tex]\( 2x \)[/tex] units, and [tex]\( x+8 \)[/tex] units, we will follow the formula for calculating the surface area of a rectangular prism. The surface area [tex]\( A \)[/tex] is given by:
[tex]\[ A = 2(lw + lh + wh) \][/tex]
Where:
- [tex]\( l \)[/tex] is the length
- [tex]\( w \)[/tex] is the width
- [tex]\( h \)[/tex] is the height
For this problem:
- [tex]\( l = x \)[/tex]
- [tex]\( w = 2x \)[/tex]
- [tex]\( h = x + 8 \)[/tex]
Now, substitute [tex]\( l \)[/tex], [tex]\( w \)[/tex], and [tex]\( h \)[/tex] into the surface area formula:
[tex]\[ A = 2 \left( (x)(2x) + (x)(x + 8) + (2x)(x + 8) \right) \][/tex]
Calculate each term inside the parentheses:
[tex]\[ (x)(2x) = 2x^2 \][/tex]
[tex]\[ (x)(x + 8) = x^2 + 8x \][/tex]
[tex]\[ (2x)(x + 8) = 2x^2 + 16x \][/tex]
Now add these results together:
[tex]\[ 2x^2 + x^2 + 8x + 2x^2 + 16x \][/tex]
Combine like terms:
[tex]\[ (2x^2 + x^2 + 2x^2) + (8x + 16x) = 5x^2 + 24x \][/tex]
So the expression inside the parentheses is now [tex]\( 5x^2 + 24x \)[/tex]. Now multiply by 2 to get the final surface area:
[tex]\[ A = 2(5x^2 + 24x) \][/tex]
Distribute the 2:
[tex]\[ A = 10x^2 + 48x \][/tex]
Thus, the expression that represents the surface area of the prism is:
[tex]\[ 10x^2 + 48x \, \text{square units} \][/tex]
Hence, the correct answer is:
[tex]\[ 10 x^2 + 48 x \, \text{square units} \][/tex]
[tex]\[ A = 2(lw + lh + wh) \][/tex]
Where:
- [tex]\( l \)[/tex] is the length
- [tex]\( w \)[/tex] is the width
- [tex]\( h \)[/tex] is the height
For this problem:
- [tex]\( l = x \)[/tex]
- [tex]\( w = 2x \)[/tex]
- [tex]\( h = x + 8 \)[/tex]
Now, substitute [tex]\( l \)[/tex], [tex]\( w \)[/tex], and [tex]\( h \)[/tex] into the surface area formula:
[tex]\[ A = 2 \left( (x)(2x) + (x)(x + 8) + (2x)(x + 8) \right) \][/tex]
Calculate each term inside the parentheses:
[tex]\[ (x)(2x) = 2x^2 \][/tex]
[tex]\[ (x)(x + 8) = x^2 + 8x \][/tex]
[tex]\[ (2x)(x + 8) = 2x^2 + 16x \][/tex]
Now add these results together:
[tex]\[ 2x^2 + x^2 + 8x + 2x^2 + 16x \][/tex]
Combine like terms:
[tex]\[ (2x^2 + x^2 + 2x^2) + (8x + 16x) = 5x^2 + 24x \][/tex]
So the expression inside the parentheses is now [tex]\( 5x^2 + 24x \)[/tex]. Now multiply by 2 to get the final surface area:
[tex]\[ A = 2(5x^2 + 24x) \][/tex]
Distribute the 2:
[tex]\[ A = 10x^2 + 48x \][/tex]
Thus, the expression that represents the surface area of the prism is:
[tex]\[ 10x^2 + 48x \, \text{square units} \][/tex]
Hence, the correct answer is:
[tex]\[ 10 x^2 + 48 x \, \text{square units} \][/tex]
