1. **Problem statement:** We have a stainless steel block shaped as a square pyramid frustum with a base side length of 8 cm, a top side length of 4 cm, and a height of 8 cm. The coordinate origin is at the center of the base. A laser beam passes through the block along the line segment PQ with points P(-3.5, 9.5, 6) and Q(-6, 16, 8). We need to find: - a) The entry and exit points of the laser beam through the block. - b) The length of the borehole (the segment inside the block). - c) The intersection point if the laser beam is along PQ with P(1, 9, 5) and Q(-1, 15, 6). --- 2. **Formulas and rules:** - The block is a frustum of a square pyramid with height $h=8$ cm. - The base square is centered at the origin in the $xy$-plane at $z=0$ with side length 8 cm, so its corners are at $x,y=\pm4$. - The top square is parallel to the base at $z=8$ cm with side length 4 cm, so its corners are at $x,y=\pm2$. - The laser beam line can be parameterized as: $$\vec{r}(t) = \vec{P} + t(\vec{Q} - \vec{P})$$ where $t \in [0,1]$ for the segment. - To find entry and exit points, find $t$ values where the line intersects the frustum boundaries. - The frustum boundaries in $z$ are between 0 and 8. - At height $z$, the square cross-section side length $s(z)$ changes linearly: $$s(z) = 8 - \frac{(8-4)}{8}z = 8 - 0.5z$$ so half side length at height $z$ is: $$h_s(z) = \frac{s(z)}{2} = 4 - 0.25z$$ - The point $(x,y,z)$ is inside the frustum if: $$|x| \leq h_s(z) \quad \text{and} \quad |y| \leq h_s(z) \quad \text{and} \quad 0 \leq z \leq 8$$ --- 3. **Part a) Find entry and exit points for $P(-3.5,9.5,6)$ and $Q(-6,16,8)$:** - Direction vector: $$\vec{d} = \vec{Q} - \vec{P} = (-6 + 3.5, 16 - 9.5, 8 - 6) = (-2.5, 6.5, 2)$$ - Parametric line: $$x(t) = -3.5 - 2.5t$$ $$y(t) = 9.5 + 6.5t$$ $$z(t) = 6 + 2t$$ - We find $t$ values where the point is on the boundary of the frustum. - Since $z$ must be between 0 and 8, solve for $t$ when $z=0$ and $z=8$: For $z=0$: $$0 = 6 + 2t \Rightarrow t = -3$$ (not in $[0,1]$ segment) For $z=8$: $$8 = 6 + 2t \Rightarrow t = 1$$ - Check if point at $t=1$ is inside the frustum: $$x(1) = -3.5 - 2.5 = -6$$ $$y(1) = 9.5 + 6.5 = 16$$ $$z(1) = 8$$ - Half side length at $z=8$ is $4 - 0.25 \times 8 = 4 - 2 = 2$. - Check if $|x(1)| \leq 2$ and $|y(1)| \leq 2$: $$|-6| = 6 > 2, \quad |16| = 16 > 2$$ So point $Q$ is outside the frustum. - We need to find $t$ where the line enters and exits the frustum by checking when $|x(t)| = h_s(z(t))$ or $|y(t)| = h_s(z(t))$ and $z(t)$ in $[0,8]$. - Define function for half side length: $$h_s(t) = 4 - 0.25 z(t) = 4 - 0.25 (6 + 2t) = 4 - 1.5 - 0.5 t = 2.5 - 0.5 t$$ - Check $|x(t)| = h_s(t)$: $$|x(t)| = |-3.5 - 2.5 t| = 2.5 - 0.5 t$$ - Solve for $t$: Case 1: $-3.5 - 2.5 t = 2.5 - 0.5 t$ $$-3.5 - 2.5 t = 2.5 - 0.5 t$$ $$-3.5 - 2.5 t + 0.5 t = 2.5$$ $$-3.5 - 2 t = 2.5$$ $$-2 t = 6$$ $$t = -3$$ (not in $[0,1]$) Case 2: $-3.5 - 2.5 t = -(2.5 - 0.5 t)$ $$-3.5 - 2.5 t = -2.5 + 0.5 t$$ $$-3.5 - 2.5 t - 0.5 t = -2.5$$ $$-3.5 - 3 t = -2.5$$ $$-3 t = 1$$ $$t = -\frac{1}{3}$$ (not in $[0,1]$) - Check $|y(t)| = h_s(t)$: $$|y(t)| = |9.5 + 6.5 t| = 2.5 - 0.5 t$$ Case 1: $9.5 + 6.5 t = 2.5 - 0.5 t$ $$9.5 + 6.5 t + 0.5 t = 2.5$$ $$9.5 + 7 t = 2.5$$ $$7 t = -7$$ $$t = -1$$ (not in $[0,1]$) Case 2: $9.5 + 6.5 t = -(2.5 - 0.5 t)$ $$9.5 + 6.5 t = -2.5 + 0.5 t$$ $$9.5 + 6.5 t - 0.5 t = -2.5$$ $$9.5 + 6 t = -2.5$$ $$6 t = -12$$ $$t = -2$$ (not in $[0,1]$) - Since no $t$ in $[0,1]$ satisfies boundary conditions, check if the segment is inside the frustum at $t=0$: At $t=0$: $$x(0) = -3.5, y(0) = 9.5, z(0) = 6$$ Half side length at $z=6$: $$h_s(6) = 4 - 0.25 \times 6 = 4 - 1.5 = 2.5$$ Check $|x(0)| = 3.5 > 2.5$, $|y(0)| = 9.5 > 2.5$ so outside. - Check if the line segment intersects the frustum at all by testing $t$ values between 0 and 1. - Since both endpoints are outside, the laser does not pass through the block. - However, the problem implies it does, so we check intersection with the planes $z=0$ and $z=8$ and the side planes. - Alternatively, solve for $t$ where $z(t) = 0$ or $z(t) = 8$ and check if $x(t), y(t)$ are inside the square at those heights. - At $z=0$, $t=-3$ (outside segment). - At $z=8$, $t=1$ (point Q), outside square. - Check intersection with side planes: The frustum sides are planes connecting edges of base and top squares. - For simplicity, approximate the frustum as the volume between $z=0$ and $z=8$ and $|x|,|y| \leq h_s(z)$. - Since no $t$ in $[0,1]$ satisfies the inside condition, the laser does not enter the block. **Conclusion:** The laser beam segment PQ does not intersect the block. --- 4. **Part b) Length of the borehole:** Since the laser does not enter the block, the borehole length is 0. --- 5. **Part c) Intersection for laser along PQ with P(1,9,5) and Q(-1,15,6):** - Direction vector: $$\vec{d} = (-1 - 1, 15 - 9, 6 - 5) = (-2, 6, 1)$$ - Parametric line: $$x(t) = 1 - 2 t$$ $$y(t) = 9 + 6 t$$ $$z(t) = 5 + t$$ - Check $t$ where $z(t) = 0$ or $8$: For $z=0$: $$0 = 5 + t \Rightarrow t = -5$$ (not in $[0,1]$) For $z=8$: $$8 = 5 + t \Rightarrow t = 3$$ (not in $[0,1]$) - Check if points at $t=0$ and $t=1$ are inside the frustum: At $t=0$: $$x=1, y=9, z=5$$ Half side length at $z=5$: $$h_s(5) = 4 - 0.25 \times 5 = 4 - 1.25 = 2.75$$ Check $|x|=1 \leq 2.75$ (inside), $|y|=9 > 2.75$ (outside) At $t=1$: $$x = 1 - 2 = -1, y = 9 + 6 = 15, z = 6$$ Half side length at $z=6$: $$h_s(6) = 2.5$$ Check $|x|=1 \leq 2.5$ (inside), $|y|=15 > 2.5$ (outside) - The segment is outside the frustum. - Check if the line intersects the frustum by solving for $t$ where $|y(t)| = h_s(z(t))$: $$|9 + 6 t| = 4 - 0.25 (5 + t) = 4 - 1.25 - 0.25 t = 2.75 - 0.25 t$$ Case 1: $9 + 6 t = 2.75 - 0.25 t$ $$9 + 6 t + 0.25 t = 2.75$$ $$9 + 6.25 t = 2.75$$ $$6.25 t = -6.25$$ $$t = -1$$ (not in $[0,1]$) Case 2: $9 + 6 t = -(2.75 - 0.25 t)$ $$9 + 6 t = -2.75 + 0.25 t$$ $$9 + 6 t - 0.25 t = -2.75$$ $$9 + 5.75 t = -2.75$$ $$5.75 t = -11.75$$ $$t \approx -2.04$$ (not in $[0,1]$) - Similarly for $x(t)$: $$|1 - 2 t| = 2.75 - 0.25 (5 + t) = 2.75 - 0.25 t$$ Case 1: $1 - 2 t = 2.75 - 0.25 t$ $$1 - 2 t + 0.25 t = 2.75$$ $$1 - 1.75 t = 2.75$$ $$-1.75 t = 1.75$$ $$t = -1$$ (not in $[0,1]$) Case 2: $1 - 2 t = -(2.75 - 0.25 t)$ $$1 - 2 t = -2.75 + 0.25 t$$ $$1 - 2 t - 0.25 t = -2.75$$ $$1 - 2.25 t = -2.75$$ $$-2.25 t = -3.75$$ $$t = \frac{5}{3} \approx 1.67$$ (not in $[0,1]$) - No intersection within segment. **Conclusion:** The laser beam segment does not intersect the block in part c either. --- **Final answers:** - a) The laser beam segment PQ with given points does not enter or exit the block. - b) The borehole length is 0. - c) The laser beam segment with new points also does not intersect the block.