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EXAMPLE 5

sin 2 θ. d =. v 2. 32. EXAMPLE 5. Calculate horizontal distance traveled. Robotics.

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EXAMPLE 5

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  1. sin 2θ d = v2 32 EXAMPLE 5 Calculate horizontal distance traveled Robotics The “frogbot” is a robot designed for exploring rough terrain on other planets. It can jump at a 45° angle and with an initial speed of 16 feet per second. On Earth, the horizontal distance d(in feet) traveled by a projectile launched at an angle θ and with an initial speed v(in feet per second) is given by: How far can the frogbot jump on Earth?

  2. 162 sin (2 45°) d = 32 sin 2θ d = v2 32 EXAMPLE 5 Calculate horizontal distance traveled SOLUTION Write model for horizontal distance. Substitute 16 for vand 45° for θ. = 8 Simplify. The frogbot can jump a horizontal distance of 8 feet on Earth.

  3. A rock climber is using a rock climbing treadmill that is 10.5 feet long. The climber begins by lying horizontally on the treadmill, which is then rotated about its midpoint by 110° so that the rock climber is climbing towards the top. If the midpoint of the treadmill is 6 feet above the ground, how high above the ground is the top of the treadmill? EXAMPLE 6 Model with a trigonometric function Rock climbing

  4. y Substitute 110° for θ and = 5.25 for r. sin 110° = 5.25 10.5 sin θ = 2 4.9 y y r EXAMPLE 6 Model with a trigonometric function SOLUTION Use definition of sine. Solve for y. The top of the treadmill is about 6 + 4.9 = 10.9 feet above the ground.

  5. 10. Estimate the horizontal distance traveled by a track and field long jumper who jumps at an angle of 20° and with an initial speed of 27 feet per second. 272 sin (2 20°) d = 32 sin 2θ d = v2 32 for Examples 5 and 6 GUIDED PRACTICE TRACK AND FIELD SOLUTION Write model for horizontal distance. Substitute 27 for vand 20° for θ. = 14.64 Simplify. The long jumper can jump 14.64 feet.

  6. y Substitute 100° for θ and = 5.25 for r. sin 100° = 5.25 10.5 sin θ = 2 5.2 y y r for Examples 5 and 6 GUIDED PRACTICE 11. WHAT IF?In Example 6, how high is the top of the rock climbing treadmill if it is rotated 100° about its midpoint? SOLUTION Use definition of sine. Solve for y. The top of the treadmill is about 6 + 5.2 = 11.2 feet above the ground.

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