Roy kayaked up the river and then back in a total of 5 hours. The trip was 4 miles each way and the current was difficult. If Roy kayaked at a speed of 4 mph, what was the speed of the current? Round your answer to 2 decimal places.
4.3
To find the speed of the current, we can use the concept of relative speed.
Let's assume the speed of the current is 'x' mph.
When Roy kayaks up the river, he needs to overcome the current, so his effective speed is (4 - x) mph.
When Roy kayaks back downstream, the current helps him, so his effective speed is (4 + x) mph.
We know that the total time taken for the trip is 5 hours.
The upstream distance and downstream distance are both 4 miles.
Using the formula Distance = Speed x Time, we can write the following equations:
4 = (4 - x) * t1 ... (1) [upstream distance]
4 = (4 + x) * t2 ... (2) [downstream distance]
We want to find the speed of the current 'x', so we need to solve these equations.
From equation (1), we can rewrite it as:
t1 = 4 / (4 - x)
From equation (2), we can rewrite it as:
t2 = 4 / (4 + x)
Since the total time taken is 5 hours, we can write:
t1 + t2 = 5
Substituting the values of t1 and t2:
4 / (4 - x) + 4 / (4 + x) = 5
Simplifying this equation, we get:
(4 + x) + (4 - x) = 5(4 - x)(4 + x)
8 = 5(16 - x^2)
8 = 80 - 5x^2
5x^2 = 72
x^2 = 72/5
x = √(72/5)
x ≈ 3.79
Therefore, the speed of the current is approximately 3.79 mph.
To find the speed of the current, we can use the formula: Speed = Distance / Time.
Let's break down the journey and calculate the time spent for each part. Roy kayaked up the river for 4 miles, which means he kayaked against the current. Since his speed is 4 mph, it took him 4 miles / 4 mph = 1 hour to complete this part.
Next, Roy kayaked back, with the current, covering the same distance of 4 miles. Here, the current assists him, so we need to factor in the speed of the current. Let's assume the speed of the current is "c" mph.
Since his speed relative to the water is the sum of his kayaking speed and the current speed (4 mph + c mph), it means that it took him 4 miles / (4 mph + c mph) = 4/(4+c) hours to kayak downstream.
Since the total time for the entire trip is 5 hours, the sum of the time spent kayaking upstream and downstream should equal 5 hours. This gives us the equation:
1 hour + 4/(4+c) hours = 5 hours.
To solve this equation, we can start by subtracting 1 hour from both sides:
4/(4+c) hours = 4 hours.
Next, we can multiply both sides of the equation by (4 + c) to eliminate the fraction:
4 = 4 hours * (4 + c).
Now, we can distribute the multiplication on the right side:
4 = 16 + 4c.
To isolate the variable, we can subtract 16 from both sides:
4 - 16 = 4c.
Simplifying further:
-12 = 4c.
Finally, we divide both sides by 4 to solve for c:
-12/4 = c.
Therefore, the speed of the current is -3 mph.
However, the speed of the current cannot be negative. Therefore, there might be an error in the problem statement. Please double-check the information provided.
let speed of current be x mph
Roy's speed with the current = 4+x
Roy's speed against the current = 4-x
time with the current = 4/(4+x)
time against current = 4/(4-x)
total time = 5
4/(4+x) + 4(4-x) = 5
multiply each term by (4+x)(4-x)
4(4-x) + 4(4+x) = 5(4-x)(4+x)
16 - 4x + 16 + 4x = 80 - 5x^2
5x^2 = 48
x^2 = 48/5
x = .....