Two ships make the same voyage of 3000 nautical miles. The faster ship travels 10 knots faster than the slower one (a knot is 1 nautical mile per hour). The faster ship makes the voyage in 50 hr less time than the slower one. Find the speeds of the two ships.
Speed of faster ship = f
Speed of slower ship = f-10
Distance = 3000 n.miles
Using distance = speed * time, which means
time = distance/speed
3000/(f-10)-3000/(f) = 50
multiply by f(f-10) on both sides:
3000f-3000(f-10) = 50(f)(f-10)
Expand and simplify:
50*f^2-500*f-30000=0
or
f^2-10*f-600=0
Factorize
(f-30)(f+20)=0
giving f=30 or f=-20. Reject negative solution of f=-20.
We are left with
f=30
The faster ship's speed is 30 knots.
So the slower ship cruises at 20 knots.
Let's denote the speed of the slower ship as x knots per hour.
Based on the problem, we know that the speed of the faster ship is 10 knots per hour faster than the slower ship. Therefore, the speed of the faster ship is (x + 10) knots per hour.
We also know that both ships traveled the same distance of 3000 nautical miles.
To find the time it took for each ship to travel this distance, we can use the formula:
Time = Distance / Speed
For the slower ship:
Time_taken_by_slower_ship = 3000 / x
For the faster ship:
Time_taken_by_faster_ship = 3000 / (x + 10)
According to the problem, the faster ship took 50 hours less time than the slower ship. Therefore, we can set up the equation:
Time_taken_by_slower_ship - Time_taken_by_faster_ship = 50
Substituting the values, we get:
3000 / x - 3000 / (x + 10) = 50
To solve this equation, we can first find a common denominator:
(3000(x + 10)) / x(x + 10) - (3000x) / x(x + 10) = 50
Simplifying further:
(3000x + 30000 - 3000x) / x(x + 10) = 50
30000 / x(x + 10) = 50
Cross-multiplying:
30000 = 50(x^2 + 10x)
Dividing both sides by 50:
600 = x^2 + 10x
Rearranging the equation:
x^2 + 10x - 600 = 0
Factoring this quadratic equation:
(x + 30)(x - 20) = 0
Setting each factor to zero:
x + 30 = 0 or x - 20 = 0
Solving each equation:
x = -30 or x = 20
Since speed cannot be negative, we discard the solution x = -30.
Therefore, the speed of the slower ship is 20 knots per hour and the speed of the faster ship is (20 + 10) = 30 knots per hour.
To find the speeds of the two ships, let's assume the slower ship's speed is x knots (nautical miles per hour). Since the faster ship is traveling 10 knots faster than the slower ship, its speed would be x + 10 knots.
Now, let's use the formula: distance = speed × time, to create two equations based on the given information.
For the slower ship:
3000 miles = x knots × t hours, where t is the time it takes for the slower ship to complete the journey.
For the faster ship:
3000 miles = (x + 10) knots × (t - 50) hours, where t - 50 is the time it takes for the faster ship to complete the journey (50 hours less than the slower ship).
Now, let's solve the provided equations to find the values of x and t, which will give us the speeds of the two ships.
Equation 1: 3000 = x × t
Equation 2: 3000 = (x + 10) × (t - 50)
Expanding the equations gives us:
Equation 1: 3000 = xt
Equation 2: 3000 = xt - 50x + 10t - 500
Combining like terms:
-50x + 10t = 500
Now, we need a second equation to solve for both x and t. Let's rearrange Equation 1 to solve for t:
t = 3000/x
Substituting this value of t into the second equation:
-50x + 10(3000/x) = 500
Expanding the equation and simplifying further:
-50x + 30000/x = 500
-50x^2 + 30000 = 500x
Bringing all terms to one side:
50x^2 + 500x - 30000 = 0
Now, let's solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a = 50, b = 500, and c = -30000.
Plugging in the values:
x = (-500 ± √(500^2 - 4 * 50 * -30000)) / (2 * 50)
Simplifying:
x = (-500 ± √(250000 + 6000000)) / 100
Calculating further:
x = (-500 ± √6250000) / 100
x = (-500 ± 2500) / 100
Now, let's consider both possible values for x:
Case 1: x = (-500 + 2500) / 100 = 20 knots
Case 2: x = (-500 - 2500) / 100 = -30 knots (discarding this value since speed cannot be negative)
Therefore, the speed of the slower ship is 20 knots. The speed of the faster ship would be 20 + 10 = 30 knots.
Hence, the speeds of the two ships are 20 knots and 30 knots, respectively.