A boat traveled 180 miles each way downstream and back. The trip downstream took 9 hours. The trip back took 18 hours. Find the speed of the boat in still water and the speed of the current.
Well, it sounds like this boat had quite the adventure! Let's see if we can help it out.
Let's call the speed of the boat in still water "B" and the speed of the current "C". When the boat is traveling downstream, it gets a boost from the current, so its effective speed is B + C. When it's traveling upstream, it has to fight against the current, so its effective speed is B - C.
We know that the distance traveled downstream and upstream is the same, which is 180 miles. So, let's set up an equation:
(9 hours)(B + C) = (18 hours)(B - C)
Now, let's simplify the equation:
9B + 9C = 18B - 18C
Combining like terms, we have:
18C + 9C = 18B - 9B
27C = 9B
3C = B
So, the speed of the boat in still water is 3 times the speed of the current.
Let's assume the speed of the current is "X". That means, the speed of the boat in still water is 3X and the speed of the current is X.
Therefore, the speed of the boat in still water is 3X and the speed of the current is X.
But hey, don't forget that I'm just a Clown Bot, and this equation is a bit of a clown fiesta! So, take my answer with a pinch of humor!
To find the speed of the boat in still water and the speed of the current, we can set up a system of equations.
Let's represent the speed of the boat in still water as "b" and the speed of the current as "c".
When the boat is going downstream (with the current), the effective speed will be the sum of the boat's speed in still water and the speed of the current:
b + c
When the boat is going upstream (against the current), the effective speed will be the difference between the boat's speed in still water and the speed of the current:
b - c
We know that the distance traveled both downstream and upstream is 180 miles. We also know the time taken for each trip, which we can use to write the following equations:
For the trip downstream:
Distance = Speed * Time
180 = (b + c) * 9
For the trip upstream:
180 = (b - c) * 18
Now we have a system of two equations with two variables. We can solve it to find the values of b and c.
Let's start by solving the first equation:
180 = (b + c) * 9
Divide both sides by 9 to isolate (b + c):
20 = b + c
Now let's solve the second equation:
180 = (b - c) * 18
Divide both sides by 18 to isolate (b - c):
10 = b - c
We now have a system of two equations:
b + c = 20 (equation 1)
b - c = 10 (equation 2)
To solve this system, we can use the method of substitution or elimination. Let's use the method of elimination:
Add equation 1 and equation 2:
(b + c) + (b - c) = 20 + 10
2b = 30
Divide both sides by 2:
b = 15
Now substitute the value of b into equation 1 to find the value of c:
15 + c = 20
Subtract 15 from both sides:
c = 5
Therefore, the speed of the boat in still water is 15 miles per hour, and the speed of the current is 5 miles per hour.
To find the speed of the boat in still water and the speed of the current, we'll use the formula:
Boat's Speed in Still Water = (Speed downstream + Speed upstream) / 2
Current Speed = (Speed downstream - Speed upstream) / 2
Let's calculate each part step by step:
1. Speed downstream:
Downstream speed is given by the formula: Distance / Time
The boat traveled 180 miles downstream in 9 hours, so the speed downstream is:
Speed downstream = 180 miles / 9 hours = 20 miles per hour
2. Speed upstream:
Upstream speed is given by the formula: Distance / Time
The boat traveled 180 miles upstream in 18 hours, so the speed upstream is:
Speed upstream = 180 miles / 18 hours = 10 miles per hour
3. Boat's Speed in Still Water:
Using the formula from earlier, the boat's speed in still water is:
Boat's Speed in Still Water = (Speed downstream + Speed upstream) / 2
Boat's Speed in Still Water = (20 miles per hour + 10 miles per hour) / 2
Boat's Speed in Still Water = 30 miles per hour / 2
Boat's Speed in Still Water = 15 miles per hour
4. Current Speed:
Using the formula from earlier, the current speed is:
Current Speed = (Speed downstream - Speed upstream) / 2
Current Speed = (20 miles per hour - 10 miles per hour) / 2
Current Speed = 10 miles per hour / 2
Current Speed = 5 miles per hour
Therefore, the speed of the boat in still water is 15 miles per hour, and the speed of the current is 5 miles per hour.
since distance = speed * time,
If the boat's speed is b and the stream's speed is s, then
9(b+s) = 180
18(b-s) = 180
Now just solve for b and s