To find the correct expression for the volume of the prism with a right triangle base and dimensions [tex]\(h = x+1\)[/tex], [tex]\(b = x\)[/tex], and [tex]\(l = x+7\)[/tex], we need to follow these steps:
1. Determine the area of the triangular base:
The base of the prism is a right triangle with legs [tex]\(b = x\)[/tex] and [tex]\(l = x+7\)[/tex]. The area of a right triangle is given by:
[tex]\[
\text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
Here, the base is [tex]\(b = x\)[/tex] and the height of the triangle (l) is [tex]\(x + 7\)[/tex]. Thus, the area of the right triangular base is:
[tex]\[
\text{Base Area} = \frac{1}{2} \times x \times (x + 7)
\][/tex]
2. Simplify the area of the base:
Simplifying the expression for the base area:
[tex]\[
\text{Base Area} = \frac{1}{2} \times x \times (x + 7) = \frac{1}{2} \times (x^2 + 7x) = \frac{1}{2}x(x + 7)
\][/tex]
3. Find the volume of the prism:
The volume [tex]\(V\)[/tex] of a prism is given by the product of the base area and the height [tex]\(h\)[/tex]:
[tex]\[
V = \text{base area} \times \text{height}
\][/tex]
Here, the height [tex]\(h\)[/tex] of the prism is [tex]\(x + 1\)[/tex], so we multiply the base area by the height:
[tex]\[
V = \left(\frac{1}{2}x(x + 7)\right) \times (x + 1)
\][/tex]
4. Expand and simplify the volume expression:
Expanding this expression:
[tex]\[
V = \frac{1}{2}x(x + 7)(x + 1)
\][/tex]
Multiplying out the terms inside the parentheses:
[tex]\[
x(x + 7)(x + 1) = x(x^2 + 8x + 7)
\][/tex]
Distributing [tex]\(x\)[/tex] through the polynomial:
[tex]\[
x(x^2 + 8x + 7) = x^3 + 8x^2 + 7x
\][/tex]
Therefore, we have:
[tex]\[
V = \frac{1}{2}(x^3 + 8x^2 + 7x)
\][/tex]
So the correct expression for the volume of the prism corresponding to the given dimensions is:
[tex]\[
\boxed{D. \quad V = \frac{1}{2}(x^3 + 8x^2 + 7x)}
\][/tex]
