1000 CHAPTER 10 Analytic Geometry 52. (Modeling) Radio Telescope Design A telescope has diameter 400 ft and maximum depth 50 ft. (a) Write an equation of a parabola that models the cross section of the dish if the vertex is placed at the origin and the parabola opens up. (b) The receiver must be placed at the focus of the parabola. How far from the vertex should the receiver be located? 53. Parabolic Arch An arch in the shape of a parabola has the dimensions shown in the figure. How wide is the arch 9 ft up? 54. Height of Bridge Cable Supports The cable in the center portion of a bridge is supported as shown in the figure to form a parabola. The center vertical cable is 10 ft high, the supports are 210 ft high, and the distance between the two supports is 400 ft. Find the height of the remaining vertical cables, if the vertical cables are evenly spaced. (Ignore the width of the supports and cables.) 55. (Modeling) Path of a Cannon Shell The physicist Galileo observed that certain projectiles follow a parabolic path. For instance, if a cannon fires a shell at a 45° angle with a speed of v feet per second, then the path of the shell (see the figure on the left below) is modeled by the following equation. y = x - 32 v2 x2 The figure on the right below shows the paths of shells all fired at the same speed but at different angles. The greatest distance is achieved with a 45° angle. The outline, or envelope, of this family of curves is another parabola with the cannon as focus. The horizontal line through the vertex of the envelope parabola is a directrix for all the other parabolas. Suppose all the shells are fired at a speed of 252.982 ft per sec. 12 ft 12 ft 400 NOT TO SCALE 210 10 –2000 –1000 1000 2000 x y 500 1000 (a) What is the greatest distance, to the nearest foot, that a shell can be fired? (b) What is the equation of the envelope parabola? (c) Can a shell reach a helicopter 1500 ft due east of the cannon flying at a height of 450 ft? 56. (Modeling) Path of a Projectile When a projected object moves under the influence of a constant force (without air resistance), its path is parabolic. This occurs when a ball is thrown near the surface of a planet or other celestial body. Suppose two balls are simultaneously thrown upward at a 45° angle on two different planets. If their initial velocities are both 30 mph, then their paths can be modeled by the following equation. y = x - g 1936 x2
RkJQdWJsaXNoZXIy NjM5ODQ=