a steamboat travels 9 miles up a river in 3 hours and returns the same distance down the river in 2 hours. find the speed of the boat in still water and the rate of the current.
speed of boat in still water --- x mph
speed of current --- y mph
distance up river = time (rate) = 3(x-y) = 9
3x-3y = 9
x-y=3
distance down river = 2(x+y) = 9
2x+2y=9
multiply first by 2
2x-2y = 6
2x+2y = 9
add them
4x = 15
x = 15/4 = 3.75 mph
then in x-y = 3
3.75-y = 3
y = .75 mph
the boat has a speed of 3.75 mph in still water, and
the current has a speed of .75 mph
check:
speed with the current = 4.5
distance = 2(4.5) = 9
speed against current = 3
distance = 3(3) = 9
The current of a river is 2 miles per hour. A boat travels to a point 8 miles uptream and back in a total of 3 hours. What is the speed of the boat in still water
To find the speed of the boat in still water and the rate of the current, we can use the formula:
Boat speed in still water = (speed upstream + speed downstream) / 2
Rate of the current = (speed downstream - speed upstream) / 2
Let's calculate the boat speed in still water first:
Speed upstream = Distance / Time = 9 miles / 3 hours = 3 miles per hour
Speed downstream = Distance / Time = 9 miles / 2 hours = 4.5 miles per hour
Boat speed in still water = (3 miles per hour + 4.5 miles per hour) / 2 = 7.5 / 2 = 3.75 miles per hour
Now, let's calculate the rate of the current:
Rate of the current = (4.5 miles per hour - 3 miles per hour) / 2 = 1.5 / 2 = 0.75 miles per hour
Therefore, the speed of the boat in still water is 3.75 miles per hour and the rate of the current is 0.75 miles per hour.
To find the speed of the boat in still water and the rate of the current, we can use the concept of relative speed.
Let's denote the speed of the boat in still water as B and the rate of the current as C.
When the boat is traveling upstream (against the current), its effective speed is reduced by the rate of the current. Thus, the speed of the boat traveling upstream can be represented as (B - C).
Similarly, when the boat is traveling downstream (with the current), its effective speed is increased by the rate of the current. So, the speed of the boat traveling downstream can be represented as (B + C).
We are given that the boat travels 9 miles upstream in 3 hours and returns the same distance downstream in 2 hours. Based on this information, we can set up the following equations:
Distance = Speed × Time
For upstream travel:
9 = (B - C) × 3 -- Equation 1
For downstream travel:
9 = (B + C) × 2 -- Equation 2
Now we have two equations with two unknowns (B and C). We can solve this system of equations to find the values of B and C.
Let's start by rearranging Equation 1:
(B - C) × 3 = 9
B - C = 9/3
B - C = 3 -- Equation 3
Next, rearrange Equation 2:
(B + C) × 2 = 9
B + C = 9/2
B + C = 4.5 -- Equation 4
Now we have a system of equations (Equations 3 and 4) that we can solve simultaneously. One way to solve this system is by adding the two equations together:
(B - C) + (B + C) = 3 + 4.5
2B = 7.5
B = 7.5 / 2
B = 3.75
Now we can substitute the value of B back into one of the equations to solve for C. Let's use Equation 3:
3.75 - C = 3
-C = 3 - 3.75
-C = -0.75
C = 0.75
Therefore, the speed of the boat in still water (B) is 3.75 miles per hour, and the rate of the current (C) is 0.75 miles per hour.