To find the volume of a prism, we multiply the area of the base by the length of the prism.
Given a right triangle as the base:
- Height [tex]\( h = x + 1 \)[/tex]
- Base [tex]\( b = x \)[/tex]
- Length [tex]\( l = x + 7 \)[/tex]
First, calculate the area of the triangular base:
[tex]\[ \text{Area of base} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times x \times (x + 1) \][/tex]
Simplify the expression for the area of the base:
[tex]\[ \text{Area of base} = \frac{1}{2} \times x \times (x + 1) = \frac{1}{2} \times (x^2 + x) = \frac{1}{2} x (x + 1) \][/tex]
Next, calculate the volume of the prism by multiplying this area by the length [tex]\( l \)[/tex]:
[tex]\[ \text{Volume} = \left(\frac{1}{2} x (x + 1)\right) \times (x + 7) \][/tex]
Expand and simplify the expression for volume:
[tex]\[ \text{Volume} = \frac{1}{2} x (x + 1)(x + 7) \][/tex]
Distribute [tex]\( (x + 1)(x + 7) \)[/tex]:
[tex]\[ (x + 1)(x + 7) = x^2 + 7x + x + 7 = x^2 + 8x + 7 \][/tex]
Now multiply by [tex]\( \frac{1}{2} x \)[/tex]:
[tex]\[ \text{Volume} = \frac{1}{2} x (x^2 + 8x + 7) = \frac{1}{2} (x^3 + 8x^2 + 7x) \][/tex]
Thus, the correct expression for the volume of the prism is:
[tex]\[ \boxed{ \frac{1}{2} \left( x^3 + 8 x^2 + 7 x \right) } \][/tex]
So, the correct answer is:
D. \(\quad V = \frac{1}{2}\left(x^3 + 8 x^2 + 7 x\right)
