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Homework answers / question archive / Consider the following scenario: Red Bird, Yellow Bird, Blue Bird and Black Bird are angry with the pigs who stole the birds’ eggs
Consider the following scenario:
Red Bird, Yellow Bird, Blue Bird and Black Bird are angry with the pigs who stole the birds’ eggs. The birds want their eggs back and will stop at nothing to get them back. The flight path of the birds can be modeled with a parabola where “x” is the distance and “y” is the height.
Use the data below to help answer the following questions:
Red Bird starts his flight from point (10, 0). His flight path reaches a maximum height of 18 yards and lands at point (38, 0).
Yellow Bird’s flight path can be modeled by the quadratic equation y=-x^2+14x-24
Blue Bird’s flight is modeled by the following graph:
(0,26)
The table below contains partial data points of Black Bird’s trajectory:
|
x |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
|
y |
0 |
7.5 |
14 |
19.5 |
24 |
27.5 |
30 |
31.5 |
32 |
31.5 |
In developing responses to the problems, be sure to show all work:
What is the maximum height of each bird’s flight:
What is the axis of symmetry for each bird’s flight:
What was the total horizontal distance of each bird’s flight:
Which bird flew the highest?
Which bird traveled the greatest horizontal distance?
Which bird hit the following pigs:
King Pig located at point (21, 19.5)
Moustache Pig located at point (9, 21)
Answer:
The parabola is a symmetric figure.
Let us first consider the flight path of the Red Bird. The Red bird takes off at the point (10,0) and lands at the point (38.0). Therefore, the line of symmetry is x = 24 and since the maximum height is 18 yards, the vertex of the parabola is ( 24,18). Let the equation of this parabola be y = a ( x-24)2 + 18 [ the vertex form of equation is y = a (x-h)2 + k where (h,k) is the vertex]. Since the Red bird passes through the point (38,0) , we have a (38-24)2 +18 or, 196a + 18 = 0 or, a = -18/196 = -9/98 . Therefore, the pathof the Red Bird's flight is the parabola y = -9/98(x-24)2 + 18 ....(1)
The flight path of the yellow bird is the parabola y = -x2 +14x -24 = - (x2 -14x + 49) +25 = - (x-7)2 + 25...(2)
The flight path's fraph of Blue Bird is not attached.
It can be ascertained from the table depicting the Black Bird's flight path that the vertex of the parabola is (16,32). Let the Black Bird's flight path be y = a (x-16)2 + 32. Since the point (8,0) is on the flight path, we have 0 = a (8- 16)2 +32 or, 64a -32 = 0 or., a = -1/2. Therefore the flight path of Black Bird is y = -1/2 (x -16)2 + 32...(3)
The vertex is the maximum point of each Bird's flight. Therefore, the maximum height of various Birds' flights are as under:
i.Red Bird- 18 yards
ii. Yellow Bird- 25 yards
iii. Blue Bird- NOT KNOWN
iv. Black Bird- 32 yards
The Axis of symmetry are as under:
i. Red Bird- x = 24
ii. Yellow Bird- x = 7
iii. Blue Bird- NOT KNOWN
iv. Black Bird- x =16
The horizontal distance of the Bird's flights are the distances between the two points where y = o and are as under:
i. Red Bird- 28 yards ( distance between ( 10,0 and (38,0)
ii. Yellow Bird- 10yards [ -(x-7)7 + 25 =0 or, (x-7) = =5 or, -5 , or, x = 2 or 12]
iii. Blue Bird- NOT KNOWN
iv. Black Bird- 16 yards ([ Takes off at (8,0) and the vertes is (16, 32) 2(16-8) = 16]
Since Blue bird's flight path is not known, we can not say which bird flew the highest. Of the other birds, the Black bird flew the highest.
Since Blue bird's flight path is not known, we can not say which bird flew the greatest horizontal distance. Of the other birds, the Red bird flew the highest horizontal distance.
The flight paths of 3 Birds are known. We can test that the points (21,19.5) and (9,21) lie on which bird's flight path.
