To find the correct expression for the volume of a prism with a right triangle base, we need to use the formula for the volume of a prism with a triangular base. For a right triangle, the volume \( V \) of the prism is given by:
[tex]\[ V = \frac{1}{2} \times \text{base} \times \text{height} \times \text{length} \][/tex]
Given the dimensions:
- Height \( h = x + 1 \)
- Base \( b = x \)
- Length \( l = x + 7 \)
We substitute these values into the formula:
[tex]\[ V = \frac{1}{2} \times x \times (x + 1) \times (x + 7) \][/tex]
To simplify this expression:
First calculate \( x \times (x + 1) \):
[tex]\[ x \times (x + 1) = x^2 + x \][/tex]
Next, multiply this result by \( (x + 7) \):
[tex]\[ (x^2 + x) \times (x + 7) = x^2(x + 7) + x(x + 7) = x^3 + 7x^2 + x^2 + 7x = x^3 + 8x^2 + 7x \][/tex]
Now, multiply the entire expression by \(\frac{1}{2}\):
[tex]\[ V = \frac{1}{2} \times (x^3 + 8x^2 + 7x) \][/tex]
So, the final simplified expression for the volume of the prism is:
[tex]\[ V = \frac{1}{2}(x^3 + 8x^2 + 7x) \][/tex]
The correct answer is:
[tex]\[ \boxed{B. \quad V=\frac{1}{2}\left(x^3+8 x^2+7 x\right)} \][/tex]
