The Gulf Stream is a warm ocean current that extends from the eastern side of the Gulf of Mexico up through the Florida Straits and along the southeastern coast of the United States to Cape Hatteras, Nort Carolina. A boat travels with the current 110 mi from Miami, Florida, to Freeport, Bahamas, in 5 hr. The return trip against the same current takes 7 6/7 hr. Find the speed of the boat in still water and the speed of the current.
See previous post.
Let's assume the speed of the boat in still water is represented by "b" and the speed of the current is represented by "c".
When the boat travels with the current from Miami to Freeport, the effective speed is the sum of the boat's speed in still water and the speed of the current. Therefore, the effective speed for this leg of the trip is b + c.
We are given that the boat travels 110 miles in 5 hours, so we can form the equation:
110 = (b + c) * 5
Similarly, when the boat travels against the current from Freeport to Miami, the effective speed is the difference between the boat's speed in still water and the speed of the current. Therefore, the effective speed for this leg of the trip is b - c.
We are given that the boat travels back 110 miles in 7 6/7 hours. To convert 7 6/7 to an improper fraction, we multiply the whole number (7) by the denominator (7) and add the numerator (6) to the product. This gives us a total of 55/7.
Now, we can form the equation for the return trip:
110 = (b - c) * (55/7)
Now, we have a system of two equations:
1) 110 = (b + c) * 5
2) 110 = (b - c) * (55/7)
To solve this system of equations, we can use the method of substitution.
From equation 1, we can isolate b + c:
b + c = 110/5
b + c = 22
Next, we substitute this value of b + c into equation 2:
110 = (b - c) * (55/7)
110 = (22 - c) * (55/7)
Now, we can cross-multiply and solve for c:
110 * 7 = 55 * (22 - c)
770 = 1210 - 55c
55c = 1210 - 770
55c = 440
c = 440/55
c = 8
Now that we have the value of c (speed of the current), we can substitute it back into equation 1 to find b (speed of the boat in still water):
b + 8 = 22
b = 22 - 8
b = 14
Therefore, the speed of the boat in still water is 14 mph, and the speed of the current is 8 mph.
To solve this problem, we need to utilize the concept of relative velocity. Let's assume the speed of the boat in still water to be "b" and the speed of the current to be "c."
During the journey from Miami to Freeport, the boat is traveling with the current, so the effective speed will be the sum of the boat speed and the current speed: (b + c).
On the return trip from Freeport to Miami, the boat is traveling against the current, so the effective speed will be the difference between the boat speed and the current speed: (b - c).
Using the formula distance = speed × time, we can set up two equations to represent the given information:
Equation 1: (b + c) × 5 = 110
Equation 2: (b - c) × (7 + 6/7) = 110
Now, let's solve these equations step by step:
For Equation 1:
(b + c) × 5 = 110
b + c = 22 (Divided both sides by 5)
For Equation 2:
(b - c) × (7 + 6/7) = 110
(b - c) × (55/7) = 110
(b - c) = 14 (Multiplied both sides by 7/55)
Now, we have a system of equations:
b + c = 22 (Equation 3)
b - c = 14 (Equation 4)
To solve these equations simultaneously, we can either eliminate "c" by adding Equation 3 and Equation 4 or find "b" by subtracting Equation 4 from Equation 3. Let's use the substitution method to find "b":
Add Equation 3 and Equation 4:
(b + c) + (b - c) = 22 + 14
2b = 36
b = 18 (Divided both sides by 2)
Now, substitute the value of "b" into Equation 3 to find "c":
18 + c = 22
c = 4
Therefore, the speed of the boat in still water is 18 mph, and the speed of the current is 4 mph.