To find the volume of a prism with a right triangle base, let's start with the general formula for the volume of a prism:
[tex]\[ V = \text{Base Area} \times \text{Height} \][/tex]
In this case, the base of the prism is a right triangle. The area [tex]\( \text{Base Area} \)[/tex] of a right triangle is given by:
[tex]\[ \text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
For the given prism, let's define:
- [tex]\( \text{base} = b = x \)[/tex]
- [tex]\( \text{height} \)[/tex] of the triangle's base = [tex]\( h = x + 1 \)[/tex]
Thus, the area of the triangular base becomes:
[tex]\[ \text{Base Area} = \frac{1}{2} \times x \times (x + 1) \][/tex]
Now, the length [tex]\( l \)[/tex] of the prism is:
[tex]\[ l = x + 7 \][/tex]
To find the volume [tex]\( V \)[/tex] of the prism, multiply the base area by the length:
[tex]\[ V = \text{Base Area} \times l \][/tex]
[tex]\[ V = \left(\frac{1}{2} \times x \times (x + 1)\right) \times (x + 7) \][/tex]
Simplify inside the parentheses first:
[tex]\[ \frac{1}{2} \times x \times (x + 1) = \frac{1}{2} \left( x^2 + x \right) \][/tex]
Now multiply that by [tex]\( x + 7 \)[/tex]:
[tex]\[ V = \frac{1}{2} \left( x^2 + x \right) \times (x + 7) \][/tex]
Distribute [tex]\( \frac{1}{2}( x^2 + x) \)[/tex] over [tex]\( (x + 7) \)[/tex]:
[tex]\[ V = \frac{1}{2} \left( x^2 \cdot (x + 7) + x \cdot (x + 7) \right) \][/tex]
[tex]\[ V = \frac{1}{2} \left( x^3 + 7x^2 + x^2 + 7x \right) \][/tex]
[tex]\[ V = \frac{1}{2} \left( x^3 + 8x^2 + 7x \right) \][/tex]
[tex]\[ V = \frac{1}{2} x^3 + 4x^2 + \frac{7}{2}x \][/tex]
Thus, the simplified expression for the volume is:
[tex]\[ V = 0.5 \times x \times (x + 1) \times (x + 7) = \frac{1}{2} x^3 + 4x^2 + \frac{7}{2} x \][/tex]
To match it with the options provided:
The correct answer is:
A. [tex]\[ V = \frac{1}{2} \left( x^3 + 8x^2 + 7x \right) \][/tex]
