Airplane A travels 1400
km at a certain speed. Plane B travels 1000 km at a speed
50 km /h faster than plane A in 3 hrs less time. Find the speed of each plane
Plane A speed = 200 mph
Plane B speed = 250 mph
i'll explain:
1400/x=1000/(x+50)+3
1400(x+50)=1000x+3x2+150x
1400x+70000=3x2+1150x
then,
3x2-250x-70000=0
(3x+350)(x-200)=0
x=200 or -350/3
Taking out the negative answer, we get the speed of plane A to be 200kph, and the speed of plane B to be 250kph
Let's assume the speed of airplane A is "x" km/h.
Airplane A travels 1400 km at a speed of "x" km/h, so it takes (1400/x) hours to complete the journey.
The speed of airplane B is 50 km/h faster than airplane A, which means the speed of airplane B is (x + 50) km/h.
Airplane B travels 1000 km at a speed of (x + 50) km/h, and it takes (1400/x - 3) hours to complete the journey.
We can set up the following equation based on the given information:
1000 / (x + 50) = 1400 / x - 3
To solve this equation, we can cross multiply:
1000x - 3000 = 1400(x + 50)
1000x - 3000 = 1400x + 70000
600x = 73000
x = 73000 / 600
x ≈ 121.67
Therefore, the speed of airplane A is approximately 121.67 km/h.
The speed of airplane B is 50 km/h faster than airplane A:
x + 50 ≈ 121.67 + 50 ≈ 171.67
Therefore, the speed of airplane B is approximately 171.67 km/h.
To find the speed of each plane, we can use the formula Distance = Speed × Time.
For Plane A:
Distance = 1400 km
Speed = x km/h (unknown)
Time = unknown
For Plane B:
Distance = 1000 km
Speed = (x + 50) km/h (50 km/h faster than Plane A)
Time = unknown - 3 hrs (Plane B takes 3 hours less than Plane A)
Now, let's solve the equation for Plane A:
1400 km = x km/h * Time (equation 1)
Since Time is unknown, let's solve the equation for Plane B:
1000 km = (x + 50) km/h * (Time - 3 hrs) (equation 2)
Now we have a system of two equations (equations 1 and 2) with two unknowns (Speed and Time).
Simplifying equation 2:
1000 km = (x + 50) km/h * (Time - 3 hrs)
1000 km = x * (Time - 3 hrs) + 50 km/h * (Time - 3 hrs)
1000 km = x * Time - 3x + 50 km/h * Time - 150 km/h
Combining like terms:
1000 km = (x + 50 km/h) * Time - (3x + 150 km/h) (equation 3)
Now, we have two equations with two unknowns:
1400 km = x km/h * Time (equation 1)
1000 km = (x + 50 km/h) * Time - (3x + 150 km/h) (equation 3)
Let's solve these equations simultaneously.
From equation 1, we can express Time in terms of Speed:
Time = 1400 km / x km/h (equation 4)
Substituting equation 4 into equation 3:
1000 km = (x + 50 km/h) * (1400 km / x km/h) - (3x + 150 km/h)
Now we can simplify and solve for x (Speed of Plane A):
1000 km = (x + 50 km/h) * (1400 km / x km/h) - (3x + 150 km/h)
1000 km = (1400(x + 50) - x(3x + 150)) / x
To further simplify this equation, we can expand the brackets:
1000 km = (1400x + 70000 - 3x^2 - 450x) / x
Combining like terms:
1000 km = (70000 - 3x^2 + 950x) / x
Multiplying both sides by x:
1000 km * x = 70000 - 3x^2 + 950x
Rearranging the equation to get a quadratic equation in standard form:
3x^2 - 950x + 70000 = 0
Now we can solve this quadratic equation for x (Speed of Plane A) using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a = 3, b = -950, and c = 70000.
Substituting the values:
x = (-(-950) ± √((-950)^2 - 4*3*70000)) / (2*3)
Simplifying:
x = (950 ± √(902500 - 840000)) / 6
x = (950 ± √62500) / 6
Now we can calculate the two possible values of x:
x1 = (950 + √62500) / 6
x2 = (950 - √62500) / 6
x1 ≈ 213.80 km/h
x2 ≈ 42.20 km/h
Since we're looking for positive speeds, we can discard x2 (42.20 km/h) as an extraneous solution. Therefore, the speed of Plane A (x) is approximately 213.80 km/h.
To find the speed of Plane B, we can substitute the value of x (213.80 km/h) into equation 1:
1400 km = 213.80 km/h * Time
Solving for Time:
Time ≈ 6.55 hrs
Now, using the equation for Plane B:
1000 km = (213.80 km/h + 50 km/h) * (6.55 hrs - 3 hrs)
Simplifying:
1000 km = 263.80 km/h * 3.55 hrs
Dividing both sides by 3.55:
263.80 km/h ≈ 281.69 km/h
Therefore, the speed of Plane B is approximately 281.69 km/h.