Accelerated Pre-Calculus



Parametric Application Problems:Name_________________________5389245876300001. A baseball is hit straight up from a height of 5 feet with an initial velocity of 80 ft/sec. a) Write parametric equations that model the height of the ball as a function of time t. b) How high is the ball after 4 seconds? c) What is the maximum height of the ball? How many seconds does it take to reach its maximum height?5665470170815002. Kevin hits a baseball at 3 feet above the ground with an initial airspeed of 150 ft/sec at an angle of 18? with the horizontal. Will the ball clear a 20-foot wall that is 400 feet away? 3. Chris and Linda warm up in the outfield by tossing softballs to each other as in the picture16002017843500 below. Find the minimum distance between the two balls and when this distance occurs.4284345121031000 4. A Ferris wheel with a 71-foot radius turns counterclockwise one revolution every 20 seconds. Tony stands at a point 90 feet to the right of the base of the wheel. At the instant Mathew is at a point 71 feet high, Tony throws a ball toward the Ferris wheel with an initial velocity of 88 ft/sec at an angle of 80? with the horizontal. Find the minimum distance between the ball and Mathew.5. Jenny, who is 5 feet tall, is standing on top of a 40-foot building. A taller building is 25 feet from this building. The taller building is 60 feet tall and 30 feet wide. How might she throw the ball so that it would land on the roof of the taller building? -1638307810500 6. Jaime and her friends find a ball toss game at the Carson Carnival. Jaime must throw the basketball from a line marked on the ground18 feet away from the target. Jaime releases the ball from a height of 6 feet with an initial velocity of 30 feet per second and an angle of 40? from the horizontal. The ball must go into a square box that is on top of two poles. The top of the box is 10 feet from the ground. If the ball has a diameter of 12 inches and the box is 24 inches wide, will the ball go into the box? Use your work and explain why or why not. If not, how should she change the way she throws the ball? 7. A baseball is hit from a height of 3 feet above the ground. It leaves the bat with an initial velocity of 152 ft/sec at an angle of elevation of 20?. A 24 foot fence is located 400 feet away from home plate. If there is an 8 mph wind blowing directly at the batter, will the ball go over the fence? If not, what is the smallest angle at which the ball can leave the bat and be a home run? Use only parametric equations to model the entire problem situation. Explain your answers. 3522345876300008. An Air Traffic Controller is monitoring the progress of two planes. When he first makes note of the plane’s positions, Plane A is 400 miles due north of the control center and Plane B is 300 miles east of the control center. A minute later, he notices that Plane A is 6 miles east of and 8 miles south of its previous position, while Plane B is 5 miles west and 7 miles north of its previous position. The planes are currently flying at the same altitude on direct routes. Will the controller need to change the flight path of one of the planes? Parametric Application Problems:Name_________________________5389245876300001. A baseball is hit straight up from a height of 5 feet with an initial velocity of 80 ft/sec. a) Write parametric equations that model the height of the ball as a function of time t.x(t) = 0 , y(t) = 80t + 5 – 16t2 b) How high is the ball after 4 seconds? y(4) = 5 + 80(4) – 16(4)2 = 69 feet c) What is the maximum height of the ball? How many seconds does it take to reach its maximum height? Using y(t), tmax = -b/(2a) = 2.5 seconds. t(2.5) = 105 feet is the maximum height reached.5665470170815002. Kevin hits a baseball at 3 feet above the ground with an initial airspeed of 150 ft/sec at an angle of 18? with the horizontal. Will the ball clear a 20-foot wall that is 400 feet away? x(t) = 150cos (18°)t and y(t) = 3 + 150sin (18°)t – 16t2 . When x = 400 = 150cos (18°)t t = 2.80289 seconds. y(2.80289) = 7.178 feet and since the wall is 20 feet tall, the ball won’t clear the wall. 3. Chris and Linda warm up in the outfield by tossing softballs to each other as in the picture16002017843500 below. Find the minimum distance between the two balls and when this distance occurs.280797091440XL = 45cos(44°)t and yL = -16t2 + 45sin(44°)t + 5XC = 78 - 41cos(39°)t and yC = -16t2 + 41sin(39°)t + 5 And, by graphing, we find this function has a minimum distance of 6.603 feet at time = 1.2056 seconds.00XL = 45cos(44°)t and yL = -16t2 + 45sin(44°)t + 5XC = 78 - 41cos(39°)t and yC = -16t2 + 41sin(39°)t + 5 And, by graphing, we find this function has a minimum distance of 6.603 feet at time = 1.2056 seconds. 4989195540385004. A Ferris wheel with a 71-foot radius turns counterclockwise one revolution every 20 seconds. Tony stands at a point 90 feet to the right of the base of the wheel. At the instant Mathew is at a point 71 feet high, Tony throws a ball toward the Ferris wheel with an initial velocity of 88 ft/sec at an angle of 80? with the horizontal. Find the minimum distance between the ball and Mathew. And this is a minimum distance when t = 2.189 seconds and that minimum distance is 3.468 feet.5. Jenny, who is 5 feet tall, is standing on top of a 40-foot building. A taller building is 25 feet from this building. The taller building is 60 feet tall and 30 feet wide. How might she throw the ball so that it would land on the roof of the taller building? Answers will vary, but let x = (v0cos θ) t and let y = (v0sin θ)t – 16t2 + 45 Then either A) play with v0 and θ until the graph contains point (k, 60), where 15 < k < 55. (Let x2 = 25 + 6t and y = 15 for a 5 second flight and the graphic is easier to see Or B) choose a point (k, 60) as described earlier and algebraically force v and θ to fit those constraints. Example: “Throw the ball from the top of her head at a 60° angle at 45 ft/sec.” -1638307810500 6. Jaime and her friends find a ball toss game at the Carson Carnival. Jaime must throw the basketball from a line marked on the ground 18 feet away from the target. Jaime releases the ball from a height of 6 feet with an initial velocity of 30 feet per second and an angle of 40? from the horizontal. The ball must go into a square box that is on top of two poles. The top of the box is 10 feet from the ground. If the ball has a diameter of 12 inches and the box is 24 inches wide, will the ball go into the box? Use your work and explain why or why not. If not, how should she change the way she throws the ball?If x1 = 30cos(40°)t and y1 = -16t2 + 30sin(40°)t + 6 and x2 = 18.5 + t/5 and y2 = 10 for 0 < t < 5, she overshoots the box. However, if she keeps the same velocity but changes the angle to 66°, the shot should be perfect. Why did I choose those values for x2??7. A baseball is hit from a height of 3 feet above the ground. It leaves the bat with an initial velocity of 152 ft/sec at an angle of elevation of 20?. A 24 foot fence is located 400 feet away from home plate. If there is an 8 mph wind blowing directly at the batter, will the ball go over the fence? If not, what is the smallest angle at which the ball can leave the bat and be a home run? Use only parametric equations to model the entire problem situation. Explain your answers.2171706032500 Let {x1 = 152cos(20°)t – 8t and y1 = -16t2 + 152sin(20°)t + 3} and let {x2 = 400 and y2 = 24 – 4.8t} for 0 < t < 5, the ball hits the fence. SO, let 152cos(θ°)t – 8t = 400 and let -16t2 + 152sin(θ)t + 3 = 24 and we have a system of 2 equations with 2 unknowns (except that sin θ and cos θ are technically not the same). However, cos θ = (400 + 8t)/(152t) and sin θ = (16t2+ 21)/(152t) and since (sin2θ + cos2 θ = 1, solving we get t = 2.99065 sec, meaning θ= 21.162°, which appears to be correct graphically. 8. An Air Traffic Controller is monitoring the progress of two planes. When he first makes note of the plane’s positions, Plane A is 400 miles due north of the control center and Plane B is 300 miles east of the control center. A minute later, he notices that Plane A is 6 miles east of and 8 miles south of its previous position, while Plane B is 5 miles west and 7 miles north of its previous position. The planes are currently flying at the same altitude on direct routes. Will the controller need to change the flight path of one of the planes? 4076703302000 Let xA = 0 + 6t and yA = 400 – 8t (400, 0) and xB = 300 – 5t and yB = 0 + 7t2255519911860001123951245235-190535750500-504824801370 and 25.879 minutes after the secondspotting, the planes are the closest they will ever get, which is 5.376 miles from each other. According to “For a commercial airliner (as the question asked), separation will?usually?be at least?3 miles laterally,?or?1,000 feet vertically. In the enroute environment – at higher operating speeds above 10,000 feet and based on the type of Radar and distance from the antennae -- a?5 mile?rule is applied laterally. This is true in?most but not all?situations.” Therefore the Air Traffic Controller doesn’t need to worry. (300, 0) ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download