Question
- Two ships started simultaneously from a port, one to the north and the other to the east. Two hours later the distance between them was 60 km. What will be the speed of the second ship if the speed of the first ship was 6 km/hr higher than the speed of the second?
Answer: Option C
Let us assume that the second ship's speed is x km/hr.
Given that the speed of the first ship is 6 km/hr higher than the second, therefore, the speed of the first ship is (x + 6) km/hr.
We know that the distance between the two ships is 60 km after 2 hours. Let us assume that the two ships met at point P after 2 hours, forming a right-angled triangle with the port as the vertex.
Let the distance travelled by the second ship be 'd' km.
Applying Pythagoras theorem, we get:
(x + 6)^2 + d^2 = (60 + d)^2On simplifying the above equation, we get:
d^2 - 60d + x^2 - 12x - 108 = 0We know that distance = speed × time.
Distance travelled by the first ship in 2 hours = (x + 6) × 2 km
Distance travelled by the second ship in 2 hours = x × 2 km
As per the problem statement, the distance between the two ships after 2 hours is 60 km. Therefore,
(x + 6) × 2)^2 + (x × 2)^2 = 60^2Simplifying the above equation, we get:
5x^2 + 24x - 576 = 0On solving the above quadratic equation, we get x = 12 km/hr (which is the speed of the second ship).
Substituting this value of x in the earlier derived equation, we get:
d^2 - 60d + 180 = 0On solving this quadratic equation, we get d = 30 km.
Hence, the speed of the second ship is 18 km/hr (which is the option C).
To summarize, the steps involved in solving the problem are:
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Let us assume that the second ship's speed is x km/hr.
Given that the speed of the first ship is 6 km/hr higher than the second, therefore, the speed of the first ship is (x + 6) km/hr.
We know that the distance between the two ships is 60 km after 2 hours. Let us assume that the two ships met at point P after 2 hours, forming a right-angled triangle with the port as the vertex.
Let the distance travelled by the second ship be 'd' km.
Applying Pythagoras theorem, we get:
(x + 6)^2 + d^2 = (60 + d)^2On simplifying the above equation, we get:
d^2 - 60d + x^2 - 12x - 108 = 0We know that distance = speed × time.
Distance travelled by the first ship in 2 hours = (x + 6) × 2 km
Distance travelled by the second ship in 2 hours = x × 2 km
As per the problem statement, the distance between the two ships after 2 hours is 60 km. Therefore,
(x + 6) × 2)^2 + (x × 2)^2 = 60^2Simplifying the above equation, we get:
5x^2 + 24x - 576 = 0On solving the above quadratic equation, we get x = 12 km/hr (which is the speed of the second ship).
Substituting this value of x in the earlier derived equation, we get:
d^2 - 60d + 180 = 0On solving this quadratic equation, we get d = 30 km.
Hence, the speed of the second ship is 18 km/hr (which is the option C).
To summarize, the steps involved in solving the problem are:
- Assume the speed of the second ship to be x km/hr.
- Use Pythagoras theorem to form an equation based on the distance between the two ships after 2 hours.
- Use the distance = speed × time formula to form another equation based on the distance between the two ships after 2 hours.
- Solve the two equations to find the value of x (which is the speed of the second ship).
- Substitute the value of x in the earlier derived equation to find the distance travelled by the second ship.
- Solve the quadratic equation to find the distance travelled by the second ship.
- Hence, the speed of the second ship is 18 km/hr.
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