Let's solve the problem step by step: ### **Step 1: Assign Coordinates** Let’s assign coordinates to points A, B, and C. - Let A be at (0, 0) - B is 400 m east of A, so B is at (400, 0) - C is 300 m south of B, so C is at (400, -300) Let D have coordinates \((x, y)\). ### **Step 2: Depths at Each Point** - Depth at A: \(Z_A = 100\) m - Depth at B: \(Z_B = 50\) m - Depth at C: \(Z_C = 80\) m ### **Step 3: Find the Equation of the Coal Seam Plane** The coal seam passes through points (0, 0, 100), (400, 0, 50), (400, -300, 80). The equation of a plane: \[ z = ax + by + c \] Substitute the values for each point: #### For A (0, 0, 100): \[ 100 = a(0) + b(0) + c \implies c = 100 \] #### For B (400, 0, 50): \[ 50 = a(400) + b(0) + 100 \implies 50 = 400a + 100 \implies 400a = -50 \implies a = -0.125 \] #### For C (400, -300, 80): \[ 80 = a(400) + b(-300) + 100 \] Plug \(a = -0.125\): \[ 80 = (-0.125)(400) - 300b + 100\\ 80 = -50 - 300b + 100\\ 80 = 50 - 300b\\ -300b = 80 - 50\\ -300b = 30\\ b = -0.1 \] So, the equation of the coal seam plane is: \[ z = -0.125x - 0.1y + 100 \] --- ### **Step 4: Find the Depth at D** Let D be at \((x_D, y_D)\). From the diagram, D appears to be to the left of A and below it, but since no measurements are given, let's assign D at \((0, -300)\) (same y as C, same x as A). Substitute \(x = 0, y = -300\): \[ z_D = -0.125(0) - 0.1(-300) + 100 \\ z_D = 0 + 30 + 100 \\ z_D = 130 \] **Depth at D = 130 m** --- ### **Step 5: Draw the Outcrop at the Same Horizon** The outcrop is where the coal seam reaches the surface (\(z = 0\)). Set \(z = 0\) and solve for \(x\) and \(y\): \[ 0 = -0.125x - 0.1y + 100\\ 0.125x + 0.1y = 100\\ \] Divide by 0.025 for easier numbers: \[ 5x + 4y = 4000 \] **This is the equation for the outcrop line on the map.** --- ### **Final Answers:** **1. Depth to coal seam at D:** \[ \boxed{130\ \text{meters}} \] **2. Equation of the outcrop (on map):** \[ 5x + 4y = 4000 \] Draw this line on the map (x is east-west, y is north-south) to show the outcrop. --- **If you need the line drawn or more explanation, let me know!**