Flood Control Tunnel Problem: Suppose that you work for a construction company that has

Chapter 10, Problem 16

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Flood Control Tunnel Problem: Suppose that you work for a construction company that has been hired to dig a drainage tunnel under a city. The tunnel will carry excess water to the other side of the city during heavy rains, thus preventing flooding (Figure 10-8f). The tunnel is to start at ground level and then slant down until it reaches a point 100 ft below the surface. Then it will go horizontally, far enough to reach the other side of the city (not shown). Your job is to analyze the slanted part of the tunnel. a. The Engineering Department has determined that the tunnel will slant downward in the direction of the vector The centerline of the tunnel starts at the point (30, 40, 0) on the surface. The measurements are in feet. Write the particular vector equation of the centerline. b. How far along the centerline must the construction crews dig to reach the end of the slanted part of the tunnel, 100 ft below ground? What are the coordinates of this endpoint? c. Construction crews must be careful when they reach a fault plane that is in the path of the slanted part of the tunnel. The Geology Department has determined that the point (60, 90, 0) is on the fault plane where it outcrops at ground level and that the vector is normal to the plane (Figure 10-8f). Find the particular equation of the plane. d. How far along the centerline of the tunnel must the construction crews dig in order to reach the fault plane? What are the coordinates of the point at which the centerline intersects the plane? How far beneath ground level is this point?