In his motorboat, Tran travels upstream at top speed to his favorite fishing spot, a distance of 120 miles, in 8 hours. Returning, he finds that the trip downstream, still at top speed, takes only 7.5 hours. Find the speed of Tran's boat and the speed of the current.
Let the speed of Tran's boat be represented by b, and the speed of the current be represented by c.
When Tran is traveling upstream, he is working against the current, so his effective speed is b - c. The distance that he travels upstream is 120 miles, and he completes the trip in 8 hours, so we can write the equation:
120 = (b - c) * 8
When Tran is traveling downstream, he is aided by the current, so his effective speed is b + c. The distance that he travels downstream is also 120 miles, and he completes the trip in 7.5 hours, so we can write the equation:
120 = (b + c) * 7.5
Simplifying both equations, we have:
8b - 8c = 120
7.5b + 7.5c = 120
Dividing the first equation by 8, we get:
b - c = 15
Rearranging this equation, we have:
b = 15 + c
Substituting this value of b into the second equation, we have:
7.5(15 + c) + 7.5c = 120
Distributing and simplifying, we have:
112.5 + 7.5c + 7.5c = 120
Combining like terms, we have:
15c = 120 - 112.5
15c = 7.5
Dividing both sides by 15, we get:
c = 7.5/15
c = 0.5
Substituting this value of c back into the equation b = 15 + c, we have:
b = 15 + 0.5
b = 15.5
Therefore, the speed of Tran's boat is 15.5 mph, and the speed of the current is 0.5 mph.
To find the speed of Tran's boat and the speed of the current, we can set up a system of equations based on the given information.
Let's represent the speed of Tran's boat as B and the speed of the current as C.
When Tran is traveling upstream, the effective speed will be B - C (as the current is opposed to his progress).
When Tran is traveling downstream, the effective speed will be B + C (as the current is pushing him along).
From the given information, we know that Tran travels 120 miles upstream in 8 hours, so we can write the equation:
120 = (B - C) * 8
Similarly, we know that Tran travels the same distance of 120 miles downstream in 7.5 hours, so we can write the equation:
120 = (B + C) * 7.5
We now have a system of two equations:
1) 120 = (B - C) * 8
2) 120 = (B + C) * 7.5
Let's solve this system of equations.
First, let's simplify equation 1:
120 = 8B - 8C
Simplifying equation 2:
120 = 7.5B + 7.5C
We can now solve this system of equations using either substitution or elimination method. Let's use the elimination method to eliminate the variable C.
Multiplying equation 1 by 7.5 and equation 2 by 8, we have:
900 = 60B - 60C
960 = 60B + 60C
Adding the two equations, the variable C gets eliminated:
1860 = 120B
Dividing both sides by 120:
B = 1860/120
B ≈ 15.5
The speed of Tran's boat is approximately 15.5 mph.
Substituting B = 15.5 into equation 2, we can solve for C:
120 = (15.5 + C) * 7.5
120 = 116.25 + 7.5C
7.5C = 120 - 116.25
7.5C ≈ 3.75
C ≈ 0.5
The speed of the current is approximately 0.5 mph.
Therefore, Tran's boat travels at a speed of approximately 15.5 mph and the current flows at a speed of approximately 0.5 mph.
To find the speed of Tran's boat and the speed of the current, let's set up some equations.
Let's assume the speed of Tran's boat is represented by "b" and the speed of the current is represented by "c".
When Tran is traveling upstream, the effective speed of the boat will be the difference between the boat's speed and the current's speed (since the current is opposing the boat's motion). So the equation for the upstream journey is:
120 miles = 8 hours * (b - c)
When Tran is traveling downstream, the effective speed of the boat will be the sum of the boat's speed and the current's speed (since the current is aiding the boat's motion). So the equation for the downstream journey is:
120 miles = 7.5 hours * (b + c)
Now we have a system of two equations with two variables. We can solve this system to find the values of b and c.
Divide both sides of the first equation by 8:
15 miles = b - c
Divide both sides of the second equation by 7.5:
16 miles = b + c
We can now solve this system of equations. Adding the two equations together eliminates the c term:
15 + 16 = b - c + b + c
31 = 2b
b = 31/2
b = 15.5 mph
Now substitute the value of b into one of the original equations to solve for c. Let's use the first equation:
15 = 15.5 - c
c = 15.5 - 15
c = 0.5 mph
Therefore, Tran's boat has a speed of 15.5 mph and the current has a speed of 0.5 mph.