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.
Vb = velocity of boat.
Vc = velocity of current.
Downstream: (Vb+Vc) * T = 110 mi.
(Vb+Vc) * 5 = 110,
Eq1: Vb+Vc = 22 mi/h.
Upstream: (Vb-Vc) * T = 110.
(Vb-Vc) * 7 6/7 = 110,
Eq2: Vb - Vc = 14 mi/h.
Add Eq1 and Eq2:
Vb + Vc = 22.
Vb - Vc = 14
Sum : 2Vb = 36,
Vb = 18 mi/h.
In Eq1, replace Vb with 18 and solve for Vc.
Yes, that is correct.
To find the speed of the boat in still water and the speed of the current, we can use the concept of relative velocity.
Let's assume the speed of the boat in still water is B mph and the speed of the current is C mph.
When the boat travels with the current from Miami to Freeport, the effective speed will be the sum of the boat's speed in still water and the speed of the current. So the effective speed is B + C mph.
Given that the boat travels 110 miles in 5 hours, we can write the equation:
110 = (B + C) * 5
Simplifying this equation, we get:
B + C = 22 (equation 1)
On the return trip, when the boat travels against the current from Freeport to Miami, the effective speed will be the difference between the boat's speed in still water and the speed of the current. So the effective speed is B - C mph.
Given that the return trip takes 7 6/7 hours, we can convert this to an improper fraction: 7 6/7 = (7 * 7 + 6) / 7 = 55/7 hours.
We can write the equation for the return trip as:
110 = (B - C) * (55/7)
Simplifying this equation, we get:
B - C = 14 (equation 2)
Now we have a system of two equations (equation 1 and equation 2) with two unknowns (B and C).
We can solve this system of equations using any method of solving simultaneous equations. Let's solve it using the substitution method.
From equation 1, we get:
B = 22 - C
Substituting this value of B into equation 2, we get:
(22 - C) - C = 14
Simplifying this equation, we get:
22 - 2C = 14
Subtracting 22 from both sides, we get:
-2C = -8
Dividing both sides by -2, we get:
C = 4
Now we know that the speed of the current is 4 mph.
Substituting this value of C into equation 1, we get:
B + 4 = 22
Subtracting 4 from both sides, we get:
B = 18
Now we know that the speed of the boat in still water is 18 mph.
Therefore, the speed of the boat in still water is 18 mph, and the speed of the current is 4 mph.