The current river flows at rate of 4mph. In order for a boat to travel 24 miles up river and then return in a total of 5 hours how fast must the boat be able to travel in still water?
if the boat's speed is s mi/hr, then since time=distance/speed, we have
24/(s-4) + 24/(s+4) = 5
s = 11.05
To solve this problem, let's break it down into smaller steps:
Step 1: Understand the given information.
We are given that the river flows at a rate of 4 mph (miles per hour). The boat needs to travel 24 miles upstream and then return. The total travel time should be 5 hours.
Step 2: Define the variables.
Let's assume the speed of the boat in still water to be x mph.
Step 3: Determine the boat's speed with the current.
When the boat is traveling upstream (against the current), the boat's effective speed will be the difference between the boat's speed in still water (x mph) and the speed of the current (4 mph). Therefore, the boat's speed when traveling upstream is (x - 4) mph.
Step 4: Calculate the time taken to travel upstream.
Time = Distance / Speed
Using this formula, the time taken to travel 24 miles upstream (against the current) is 24 / (x - 4) hours.
Step 5: Determine the boat's speed against the current.
When the boat is traveling downstream (with the current), the boat's effective speed will be the sum of the boat's speed in still water (x mph) and the speed of the current (4 mph). Therefore, the boat's speed when traveling downstream is (x + 4) mph.
Step 6: Calculate the time taken to travel downstream.
Again, using the time = distance / speed formula, the time taken to travel 24 miles downstream (with the current) is 24 / (x + 4) hours.
Step 7: Set up the equation.
We are given that the total travel time should be 5 hours. So we can set up the equation:
Time upstream + Time downstream = 5
Or,
24 / (x - 4) + 24 / (x + 4) = 5
Step 8: Solve the equation.
Now, we need to solve the equation to find the value of x, the boat's speed in still water.
Step 9: Simplify the equation.
To simplify the equation, we multiply through by the common denominator, (x - 4)(x + 4), which gives us:
24(x + 4) + 24(x - 4) = 5(x - 4)(x + 4)
Step 10: Solve for x.
Now we can solve the resulting quadratic equation for x.