1. **State the problem:** Find the volume of the pyramid with base on the plane $z = -11$ and sides formed by the planes $y = 0$, $y - x = 1$, and $x + y + z = 0$. 2. **Understand the geometry:** The base is the polygon formed by the intersection of the planes $z = -11$, $y = 0$, and $y - x = 1$. The apex is where the three side planes intersect. 3. **Find the apex:** Solve the system: $$y = 0$$ $$y - x = 1$$ $$x + y + z = 0$$ From $y=0$, substitute into $y - x = 1$ gives $0 - x = 1 \Rightarrow x = -1$. Substitute $x = -1$, $y=0$ into $x + y + z = 0$ gives $-1 + 0 + z = 0 \Rightarrow z = 1$. So apex is at $(-1, 0, 1)$. 4. **Find the base polygon vertices:** The base lies on $z = -11$ and is bounded by $y=0$ and $y - x = 1$. - Intersection of $y=0$ and $z=-11$ is the line $y=0, z=-11$. - Intersection of $y - x = 1$ and $z=-11$ is the line $y = x + 1, z=-11$. Find intersection of $y=0$ and $y - x = 1$: Set $y=0$ in $y - x = 1$ gives $0 - x = 1 \Rightarrow x = -1$. So point $(-1, 0, -11)$. Find intersection of $y=0$ and $x + y + z = 0$ at $z=-11$: At $z=-11$, $x + y -11 = 0 \Rightarrow x + y = 11$. With $y=0$, $x=11$. So point $(11, 0, -11)$. Find intersection of $y - x = 1$ and $x + y + z = 0$ at $z=-11$: At $z=-11$, $x + y - 11 = 0 \Rightarrow x + y = 11$. From $y - x = 1$, $y = x + 1$. Substitute into $x + y = 11$: $$x + (x + 1) = 11 \Rightarrow 2x + 1 = 11 \Rightarrow 2x = 10 \Rightarrow x = 5$$ Then $y = 5 + 1 = 6$. So point $(5, 6, -11)$. 5. **Base vertices are:** $A(-1, 0, -11)$, $B(11, 0, -11)$, $C(5, 6, -11)$. 6. **Calculate base area:** Use the formula for area of triangle with vertices $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$: $$\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$$ Using $x,y$ coordinates of base points: $$= \frac{1}{2} |-1(0 - 6) + 11(6 - 0) + 5(0 - 0)|$$ $$= \frac{1}{2} | -1(-6) + 11(6) + 0 | = \frac{1}{2} |6 + 66| = \frac{1}{2} \times 72 = 36$$ 7. **Calculate height:** Height is the perpendicular distance from apex $(-1,0,1)$ to the base plane $z = -11$. Height $h = |1 - (-11)| = 12$. 8. **Calculate volume:** Volume of pyramid is $$V = \frac{1}{3} \times \text{base area} \times \text{height} = \frac{1}{3} \times 36 \times 12 = 144$$ **Final answer:** $$V = 144$$