A boat's crew rowed 10.5 miles downstream, with the current, in 1.5 hours. The return trip upstream, against the current, covered the same distance, but took 3.5 hours. Find the crew's rowing rate in still water, and the rate of the current.
(Vb+Vc)1.5 = 10.5
Eq1: Vb + Vc = 7
(Vb-Vc)3.5 = 10.5
Eq2: Vb - Vc = 3
Add Eq1 and Eq2:
Vb + Vc = 7
Vb - Vc = 3
2Vb = 10
Vb = 5 mi/h
5 + Vc = 7
Vc = 2 mi/h.
To find the crew's rowing rate in still water and the rate of the current, we can set up a system of equations.
Let's denote the crew's rowing rate in still water as "x" and the rate of the current as "y".
Downstream:
The distance covered is 10.5 miles and the time taken is 1.5 hours. When rowing downstream, the crew's effective speed is increased by the rate of the current. So the equation for the downstream trip is:
10.5 = (x + y) * 1.5
Upstream:
The distance covered is still 10.5 miles, but the time taken is now 3.5 hours. When rowing upstream, the crew's effective speed is decreased by the rate of the current. So the equation for the upstream trip is:
10.5 = (x - y) * 3.5
Now we have a system of two equations:
1) 10.5 = (x + y) * 1.5
2) 10.5 = (x - y) * 3.5
To solve this system, we can begin by simplifying the equations:
1) 10.5 = 1.5x + 1.5y
2) 10.5 = 3.5x - 3.5y
Next, we can rearrange equation 1 to solve for x in terms of y:
1) 10.5 - 1.5y = 1.5x
Divide both sides by 1.5:
(10.5 - 1.5y) / 1.5 = x
Simplify:
7 - y = x
Now we substitute this value of x in equation 2:
2) 10.5 = 3.5(7 - y) - 3.5y
Distribute:
10.5 = 24.5 - 3.5y - 3.5y
Combine like terms:
10.5 = 24.5 - 7y
Move the constant to the other side of the equation:
7y = 24.5 - 10.5
7y = 14
Divide both sides by 7:
y = 2
Now that we have the value of y, we can substitute it back into equation 1 to find x:
1) 10.5 = 1.5x + 1.5(2)
Simplify:
10.5 = 1.5x + 3
Subtract 3 from both sides:
7.5 = 1.5x
Divide both sides by 1.5:
x = 5
Therefore, the crew's rowing rate in still water is 5 miles per hour, and the rate of the current is 2 miles per hour.