EXAMPLE 5 Calculate horizontal distance traveled Robotics

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Presentation transcript:

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: d = v2 32 sin 2θ How far can the frogbot jump on Earth?

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

EXAMPLE 6 Model with a trigonometric function 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? Rock climbing

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

The long jumper can jump 14.64 feet. GUIDED PRACTICE for Examples 5 and 6 TRACK AND FIELD 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. SOLUTION d = v2 32 sin 2θ Write model for horizontal distance. d = 272 32 sin (2 20°) Substitute 27 for v and 20° for θ. = 14.64 Simplify. The long jumper can jump 14.64 feet.

GUIDED PRACTICE for Examples 5 and 6 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 sin θ = y r Use definition of sine. sin 100° = y 5.25 Substitute 100° for θ and = 5.25 for r. 2 10.5 5.2 y Solve for y. The top of the treadmill is about 6 + 5.2 = 11.2 feet above the ground.