A boat travel 4 miles upstream with a current of 5 miles per hour. It takes 40 mminutes longer to go upstream. How fast does the boat go downstream
They probably expect you to assume that the boat is using the same power to go through the water in both directions.
Let V be the boat speed relative to the water in both directions. Relative to the shore, its speed is V+5 going downstream and V-5 going upstream.
Let T be the time that it takes to go downstream, in hours. It take T + 2/3 to go upstream. You can write two equations in the two unknowns, V and T.
4 = (V+5) T
4 = (V-5)(T + 2/3)
4 = (V-5)[4/(V+5) + (2/3)]
4(V+5) = 4(V-5) + (2/3)(V^2-25)
40 = (2/3)(V^2-25)
V^2-25 = 60
V^2 = 85
V = 9.22 mph.
Downstream speed = 14.22 mph
Upstream speed = 4.22 mph
Dowstream travel time = 4/14.22
= 0.281 hrs = 16.9 minutes
Upstream travel time = 4/4.22 = 0.948 hr = 56.9 min
To find the speed of the boat downstream, we need to analyze the given information. Let's break down the problem and set up the equations.
Let's assume the speed of the boat in still water is 'x' miles per hour.
When the boat travels upstream, it has to fight against the current, which is 5 miles per hour. So the effective speed of the boat upstream will be 'x - 5' miles per hour.
Distance traveled upstream = 4 miles
Time taken to go upstream = 40 minutes longer than the time taken to go downstream
Now, let's set up the equations:
Time taken to go upstream = Distance / Speed
Time taken to go downstream = Distance / Speed
Given that the time taken to go upstream is 40 minutes longer, we can write:
Time taken to go upstream = Time taken to go downstream + 40 minutes
Converting the time taken to go upstream from minutes to hours:
Time taken to go upstream = (40/60) hours = 2/3 hours
Now, let's substitute the values into the equations:
4 / (x - 5) = 2/3 + 4 / (x)
To solve this equation, we can cross-multiply and simplify:
(4 / (x - 5)) - (4 / x) = 2/3
[(4x - 20 - 4x) / (x(x - 5))] = 2/3
[-20 / (x(x - 5))] = 2/3
Cross-multiplying:
3 * (-20) = 2 * x * (x - 5)
-60 = 2x^2 - 10x
Rearranging the equation:
2x^2 - 10x - 60 = 0
Now, we can solve this quadratic equation to find the value of x. We can either factorize, complete the square, or use the quadratic formula.
Using the quadratic formula:
x = (-(-10) ± √((-10)^2 - 4 * 2 * -60)) / (2 * 2)
x = (10 ± √(100 + 480)) / 4
x = (10 ± √580) / 4
Therefore, the boat's speed in still water, x, is approximately 9.82 mph (rounded to two decimal places) or x = 0.41 mph.
So, the boat would go downstream at a speed of 9.82 + 5 = 14.82 mph (rounded to two decimal places).
Note: It's important to double-check the equation and ensure no mistakes were made during the calculations.