4) A pyrotechnic rocket is fired from a platform 2 ft high at an angle of 30° from the horizontal with an initial speed of 72 ft/sec. Choose a coordinate system with the origin at ground level directly below the launch position. (a) Write parametric equations that model the path of the shell as a function of the time t (in sec) after launch. (b) Approximate the time required for the shell to hit the ground. Round to the nearest hundredth of a second.
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- Tony hits a baseball at a height of 3 ft from the ground. The ball leaves his bat traveling with an initial speed of 120 ft/sec at an angle of 45° from the horizontal. Choose a coordinate system with the origin at ground level directly under the point where the ball is struck. Write parametric equations that model the path of the ball as a function of time t (in sec). When is the ball at its maximum height? Give the exact value and round to the nearest hundredth of a second. What is the maximum height? If an outfielder catches the ball at a height of 6 ft, for how long was the ball in the air after being struck? Give the exact answer and the answer rounded to the nearest hundredth of a second. How far is the outfielder from home plate when he catches the ball? Round to the nearest foot.An object moves according to the parametric equations: x=sqrt(1+10t) y=t-t^5 Round each of the following to 2 decimal places. 1. Calculate the object's velocity after 2 seconds. Magnitude= Velocity direction= 2. Calculate the object's acceleration after 2 seconds. Magnitude = Acceleration direction =A particle moves from point A = (-1, –4) to point B = (-9, -6) in 2 hours at a constant rate. The coordinates are given in inches with respect the the standard xy-coordinate plane. Find the parametric equations with respect to time for the motion of the particle.
- A very tall light standard is swaying in an east-west direction in a strong wind. An observer notes that the time difference between the vertical position and the furthest point of sway was 2 seconds. The pole is 40 metres tall. At the furthest point of sway, the tip of the pole is 1° out of the vertical position when measured from the bottom of the pole. Create a sinusoidal equation that models the motion of the tip of the pole as a displacement from the vertical position as a sinusoidal function of time. Assume time starts when the tip of the pole is furthest east. Include a sketch of the graph of your equation. (4 marks)The path of a projectile that is launched h feet above the ground with an initial velocity of v0 feet per second and at an angle θ with the horizontal is given by the parametric equations x = (v0 cos θ)t and y = h + (v0 sin θ)t - 16t2,where t is the time, in seconds, after the projectile was launched.A football player throws a football with an initial velocity of100 feet per second at an angle of 40° to the horizontal. The ball leaves the player’s hand at a height of 6 feet.a. Find the parametric equations that describe the position of the ball as a function of time.b. Describe the ball’s position after 1, 2, and 3 seconds.Round to the nearest tenth of a foot.c. How long, to the nearest tenth of a second, is the ball in flight? What is the total horizontal distance that it travels before it lands?d. Graph the parametric equations in part (a) using a graphing utility. Use the graph to determine when the ball is at its maximum height. What is its maximum height?Round answers to the…Find parametric equations for the line through the point (6,7,5) and parallel to the line: x-3/-2 = y+8/7 = z-4/5
- A ball is thrown with an initial velocity of 70 km per second, at an angleof 35° with the horizontal. Find the vertical and horizontal componentsof the velocity.The maximum height of a Ferris wheel above the ground is 36 metres. The wheel takes 4 minutes to make one complete revolution. Passengers board the Ferris wheel 6 metres above the ground at the bottom of its rotation. a) Plot a rough sketch of the sinusoidal function relating the height of the passenger, h(t), to the time in seconds, t. (b) Determine the equation for h(t). c) How high is the passenger after 35 seconds? d) At what time(s) in the first rotation is the passenger at a height of 26 m.A plane directly above Denver, Colorado, (altitude 1650 meters) flies to Bismark, North Dakota (altitude 550 meters). It travels at 750 km/hour at a constant height of 7000 meters above the line joining Denver and Bismark. Bismark is about 850 km in the direction 60° north of east from Denver. Find parametric equations describing the plane's motion. Assume the origin is at sea level beneath Denver, that the x-axis points east and the y-axis points north, and that the earth is flat. Measure distances in kilometers and time in hours. r(t) = ?
- A plane directly above Denver, Colorado, (altitude 1650 meters) flies to Bismark, North Dakota (altitude 550 meters). It travels at 650 km/hour along a line at 8500 meters above the line joining Denver and Bismark. Bismark is about 850 km in the direction 60° north of east from Denver. Find parametric equations describing the plane's motion. Assume the origin is at sea level beneath Denver, that the x-axis points east and the y-axis points north, and that the earth is flat. Measure distances in kilometers and time in hours. F(t) = 10.15 k + tFind parametric equations for the line tangent to the helixWhat is the equation of the curve described by the parametric equations x = 3+cost, y = -5+sint ?