The speed of a boat in still water is 20 mph. It travels 24 mi upstream and 24 mi downstream in a total time of 10 hr. what is the speed if the current?
Recall that speed is distance traveled over time:
v = d/t
Represent the unknowns:
Let x = speed of current
Let t = time upstream
Let 10-t = time downstream (since the total time is 10 hours according to the problem)
Note that when the boat travels upstream, the current is against it, so the speed of boat is decreased), while if it travels downstream, the current has the same direction of flow as the boat, so its speed is increased.
The distance traveled in upstream and downstream are the same. We can now write the equations for distance traveled for upstream and downstream:
upstream: 24 = (20 - x)t
downstream : 24 = (20 + x)(10 - t)
You have two equations and two unknowns, so you can now solve for the unknowns. You can use substitution method to find x and t.
hope this helps~ `u`
To find the speed of the current, we can set up a system of equations based on the given information.
Let's assume the speed of the current is 'x' mph.
When the boat is traveling upstream (against the current), its effective speed is reduced by the current's speed. So, the speed of the boat upstream is (20 - x) mph.
When the boat is traveling downstream (with the current), its effective speed is increased by the current's speed. So, the speed of the boat downstream is (20 + x) mph.
Using the formula: Distance = Speed × Time, we can write the equations:
24 miles = (20 - x) mph × time upstream
24 miles = (20 + x) mph × time downstream
We are also given that the total time for both the upstream and downstream trips is 10 hours. So, we can write the equation:
time upstream + time downstream = 10 hours
Now, let's solve this system of equations step by step.
From the distance equation:
24 = (20 - x) × time upstream (Equation 1)
24 = (20 + x) × time downstream (Equation 2)
We also know that time upstream + time downstream = 10 hours:
time upstream + time downstream = 10 (Equation 3)
To solve the system of equations, we need to express time upstream and time downstream in terms of x.
Using the formula: Distance = Speed × Time, we can rearrange Equation 1 and Equation 2 to find time upstream and time downstream, respectively.
time upstream = 24 / (20 - x) (Equation 4)
time downstream = 24 / (20 + x) (Equation 5)
Now, substitute Equation 4 and Equation 5 into Equation 3 and solve for x.
24 / (20 - x) + 24 / (20 + x) = 10
To simplify this equation, we can multiply throughout by (20 - x)(20 + x) to eliminate the denominators:
24(20 + x) + 24(20 - x) = 10(20 - x)(20 + x)
Now, distribute and combine like terms:
480 + 24x + 480 - 24x = 10(400 - x^2)
960 = 4000 - 10x^2
Rearrange the equation to make it a quadratic equation:
10x^2 = 4000 - 960
10x^2 = 3040
Divide both sides by 10:
x^2 = 304
Take the square root of both sides:
x ≈ ±17.46
Since the speed of the current cannot be negative, we discard the negative value.
Therefore, the speed of the current is approximately 17.46 mph.