a patrol boat took 2.5 hours for a round trip 12km upriver and 12km back. the speed of the current was 2 km/h. what was the speed of the boat in still water?
ps: i just need to know how to make equations from the question above. it's supposed to create equation, then use the quadratic formula to solve and find the answer.
thank you!
speed of boat in still water ---- x km/h
speed of current ----- 2 km/h
speed against current = x-2
speed with the current = x+2
time upriver = 12/(x-2)
time downriver = 12/(x+2)
equation:
12/(x-2) + 12/(x+2) = 2.5
hint: multiply both sides by (x+2)(x-2), simplify and you have your quadratic
To solve this problem using equations and the quadratic formula, we need to define some variables.
Let's assume the speed of the boat in still water is represented by 'b' in km/h (kilometers per hour).
We can also assign the speed of the current as 'c,' which is given as 2 km/h.
Now, let's break down the information provided in the question:
1. The boat travels 12 km upriver. The speed of the boat relative to the river's flow is reduced by the speed of the current, so the effective speed is (b - c) km/h.
2. The boat then covers the same 12 km distance back downstream. This time, the speed of the boat relative to the river's flow is increased by the speed of the current, resulting in an effective speed of (b + c) km/h.
3. The total round trip took 2.5 hours.
Using the formula: time = distance / speed, we can express the time taken for each segment of the trip in terms of the variables:
Time taken to go upriver: 12 / (b - c) hours
Time taken to go downstream: 12 / (b + c) hours
Since the total time taken for the round trip is 2.5 hours, we can create the following equation:
12 / (b - c) + 12 / (b + c) = 2.5
Now we have our equation. To solve it using the quadratic formula, we need to rearrange it first:
[12(b + c) + 12(b - c)] / ((b - c)(b + c)) = 2.5
Simplifying further:
[(24b) / (b^2 - c^2)] = 2.5
Now, to remove the fraction, cross multiply:
24b = 2.5(b^2 - c^2)
Expanding:
24b = 2.5b^2 - 2.5c^2
Finally, rearrange it into a quadratic equation form:
2.5b^2 - 24b - 2.5c^2 = 0
Now, you can use the quadratic formula (b = (-b ± √(b^2 - 4ac)) / 2a) to solve the equation and find the value of 'b,' which represents the speed of the boat in still water.
To solve this problem using equations, let's define the variables:
Let 's' represent the speed of the boat in still water (in km/h).
Let 'c' represent the speed of the current (in km/h).
Given:
The round trip upriver and back is 12 km each way.
The speed of the current is 2 km/h.
The total time it took for the round trip was 2.5 hours.
Now, let's break down the problem into two parts: the trip upriver and the trip downstream.
1. Trip upriver:
When the boat is going upriver, it is against the current. The effective speed will be the boat's speed in still water 's' minus the speed of the current 'c'.
So, the speed of the boat upriver is: (s - c) km/h.
The distance traveled upriver is 12 km.
Using the formula: Time = Distance / Speed, we can write the equation:
Time upriver = 12 km / (s - c) km/h
2. Trip downstream:
When the boat is going downstream, it is with the current. The effective speed will be the boat's speed in still water 's' plus the speed of the current 'c'.
So, the speed of the boat downstream is: (s + c) km/h.
The distance traveled downstream is also 12 km.
Using the formula: Time = Distance / Speed, we can write the equation:
Time downstream = 12 km / (s + c) km/h
According to the given information, the total time taken for the round trip is 2.5 hours. So:
Time upriver + Time downstream = 2.5 hours
Now, substitute the values into the equation:
12 / (s - c) + 12 / (s + c) = 2.5
To solve this quadratic equation, let's multiply through by (s - c)(s + c) to eliminate the denominators:
12(s + c) + 12(s - c) = 2.5(s - c)(s + c)
Simplify the equation:
12s + 12c + 12s - 12c = 2.5(s^2 - c^2)
Combine like terms:
24s = 2.5s^2 - 2.5c^2
Rearrange the equation:
2.5s^2 - 24s - 2.5c^2 = 0
Now you can use the quadratic formula to solve for 's':
s = (-b ± √(b^2 - 4ac)) / (2a)
In this case, 'a' is 2.5, 'b' is -24, and 'c' depends on the value of the current speed. Once you have 's', you will have the speed of the boat in still water.