Quadratic Models
a real-life phenomenon that can be
represented graphically using a quadratic
equation (parabola)
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A bridge is built with an archway that is 8 feet wide at the base. Its shape can be represented by a parabola with the equation y = ­2x2 + 16x, where y is the height of the arch. Graph the parabola on the interval 0 ≤ x ≤ 8. a) Determine the maximum height, y, of the arch. b) Why might someone be concerned about the maximum height of the archway?
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A toy rocket is launched at 19.6 meters per second (m/s) from a 58.8­meter tall platform. The equation for the rocket's height y at time x seconds after launch is y = –4.9x2 + 19.6x + 58.8, where y is in meters. a) Sketch a graph of the rocket's path to the ground.
b) What is the maximum height, y, of the rocket? c) When does the rocket hit the ground? 4
Jakob and Gavin are in a car at the top of a roller­coaster ride. The distance, y, of the car from the ground as the car descends is determined by the equation y = 144 ­ 16t2 where t is the number of seconds it takes the car to travel down to each point on the ride. How many seconds will it take Jakob and Gavin to reach the ground? (Hint: What are you trying to find? When does y = on the ground?) 5
A small rocket is launched from a height of 72 feet. The height of the rocket in feet, h, is represented by the equation h(t) = ­16t2 + 64t + 72, where t = time in seconds. a) Find how many seconds until the rocket reaches its maximum height.
b) Find how many feet high it will be at that point.
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The height of a golf ball hit into the air is modeled by the equation h = ­16t2 + 96t where h represents the height, in feet, and t represents the number of seconds that have passed since the ball was hit. a) How many seconds until the ball reaches its maximum height?
b) What is its maximum height?
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An architect is designing a museum entranceway in the shape of a parabolic arch represented by the equation y = ­x2 + 20x, where 0 ≤ x ≤ 20 and all dimensions are expressed in feet. Determine its maximum height, in feet.
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A ball is launched directly upward from a platform 80 feet high. The equation for the ball's height y at time x seconds after launch is y = –16x2 + 64x + 80, where y is in feet. a) Sketch a graph of the ball's path.
b) What will be the ball's maximum height? c) When will the ball attain this height? 10
Cloe tosses a coin off a bridge into the stream below. The distance in feet, y, that the coin is above the water is modeled by the equation y = ­16x2 + 96x + 112, where x represents time in seconds.
a) What is the greatest height of the coin?
b) How much time will it take for the coin to hit the water?
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A firework is fired vertically into the air from the ground, and its path is represented by the equation y = ­16x2 + 80x. Find the highest height, y, reached by the firework just as it explodes. 100
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This highest point is located 100 feet above the point of release. 12
The profits for Mr. Walker's company can be represented by the equation y = ­3x2 + 18x ­ 4, where y is the amount of profit in hundreds of thousands of dollars and x is the number of years of operation. He realizes his company is on the downturn and wishes to sell before he ends up in debt.
a) When will Mr. Walker's business show the maximum profit?
b) What is the maximum profit?
c) At what time will it be too late to sell his business? (When will he start losing money?)
a) 3 years
b) $2,300,000
c) after 3 years
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