Question
If y^2 -py/2-1=0. Find p
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Answer
If y^2 -py/2-1=0. Find other root
Answer
Two places x (56degrees N, 45 degrees E) and y (56degrees N, 32 degrees W) if it takes an aeroplane 6 hours to fly from x to y. Calculate it's speed, correct to the nearest kilometre per hour. (Radius of the earth = 6 40 km, pie = 22/7)
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We can solve this quadratic equation by using the quadratic formula:
For a quadratic equation of the form ax^2 + bx + c = 0, the solutions x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, the equation is y^2 - py/2 - 1 = 0. So, a = 1, b = -p/2, and c = -1.
Using the quadratic formula, we have:
y = (-(-p/2) ± √((-p/2)^2 - 4(1)(-1))) / (2(1))
Simplifying the equation, we get:
y = (p/2 ± √(p^2/4 + 4)) / 2
This can be further simplified to:
y = (p ± √(p^2 + 16)) / 4
Therefore, the value of p is not a fixed number. It can take on any value, and the solutions for y will depend on the value of p.
For a quadratic equation of the form ax^2 + bx + c = 0, the solutions x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, the equation is y^2 - py/2 - 1 = 0. So, a = 1, b = -p/2, and c = -1.
Using the quadratic formula, we have:
y = (-(-p/2) ± √((-p/2)^2 - 4(1)(-1))) / (2(1))
Simplifying the equation, we get:
y = (p/2 ± √(p^2/4 + 4)) / 2
This can be further simplified to:
y = (p ± √(p^2 + 16)) / 4
Therefore, the value of p is not a fixed number. It can take on any value, and the solutions for y will depend on the value of p.
Answered by
Bot
To find the other root, we can apply the quadratic formula as we did before:
y = (p ± √(p^2 + 16)) / 4
To find the other root, we can subtract the root we found from a fixed value, such as p:
Other root = p - (p ± √(p^2 + 16)) / 4
Simplifying this expression, we get:
Other root = (4p - (p ± √(p^2 + 16))) / 4
Expanding and simplifying further, we have:
Other root = (3p ± √(p^2 + 16)) / 4
Therefore, the other root is given by (3p ± √(p^2 + 16)) / 4.
y = (p ± √(p^2 + 16)) / 4
To find the other root, we can subtract the root we found from a fixed value, such as p:
Other root = p - (p ± √(p^2 + 16)) / 4
Simplifying this expression, we get:
Other root = (4p - (p ± √(p^2 + 16))) / 4
Expanding and simplifying further, we have:
Other root = (3p ± √(p^2 + 16)) / 4
Therefore, the other root is given by (3p ± √(p^2 + 16)) / 4.
Answered by
Bot
To calculate the speed of the airplane, we need to find the distance between points x and y and divide it by the time it takes to fly from x to y.
First, we need to find the distance (arc length) between the two points on the surface of the Earth.
The distance (arc length) can be calculated using the formula:
Distance = radius of the Earth * angle subtended * π / 180
For point x:
Distance_x = (6400 km) * (45°E - 0°) * (22/7) / 180°
Distance_x = (6400 km) * (45°) * (22/7) / 180°
Distance_x = (6400 km) * (45) * (22/7) / 180
Distance_x ≈ 1803.43 km
For point y:
Distance_y = (6400 km) * (32°W - 0°) * (22/7) / 180°
Distance_y = (6400 km) * (32°) * (22/7) / 180°
Distance_y = (6400 km) * (32) * (22/7) / 180
Distance_y ≈ 1232.06 km
Now, we can find the total distance between x and y:
Total Distance = Distance_x + Distance_y
Total Distance ≈ 1803.43 km + 1232.06 km
Total Distance ≈ 3035.49 km
Finally, we can calculate the speed of the airplane:
Speed = Total Distance / Time
Speed = 3035.49 km / 6 hours
Speed ≈ 505.91 km/h
Therefore, the speed of the airplane is approximately 505.91 km/h.
First, we need to find the distance (arc length) between the two points on the surface of the Earth.
The distance (arc length) can be calculated using the formula:
Distance = radius of the Earth * angle subtended * π / 180
For point x:
Distance_x = (6400 km) * (45°E - 0°) * (22/7) / 180°
Distance_x = (6400 km) * (45°) * (22/7) / 180°
Distance_x = (6400 km) * (45) * (22/7) / 180
Distance_x ≈ 1803.43 km
For point y:
Distance_y = (6400 km) * (32°W - 0°) * (22/7) / 180°
Distance_y = (6400 km) * (32°) * (22/7) / 180°
Distance_y = (6400 km) * (32) * (22/7) / 180
Distance_y ≈ 1232.06 km
Now, we can find the total distance between x and y:
Total Distance = Distance_x + Distance_y
Total Distance ≈ 1803.43 km + 1232.06 km
Total Distance ≈ 3035.49 km
Finally, we can calculate the speed of the airplane:
Speed = Total Distance / Time
Speed = 3035.49 km / 6 hours
Speed ≈ 505.91 km/h
Therefore, the speed of the airplane is approximately 505.91 km/h.
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