It took a boat 6 h to travel 84 km up a river against the current but only 4 h and 12 minutes for the return trip with the current. Find the speed of the boat in still water and the speed of the current water.
Vb - Vw = 84km/6h = 14km/h.
Vb + Vw = 84km/4.2h = 20 km/h.
Eq1: Vb - Vw = 14
Eq2: Vb + Vw = 20
Add the Eqs:
2Vb = 34
Vb = 17 km/h. = Velocity of boat.
In Eq2, substitute 17 for Vb:
17 + Vw = 20
Vw = 3 km/h = Velocity of the water.
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 assume the speed of the boat in still water is "b" km/h and the speed of the current is "c" km/h.
When traveling upstream (against the current), the effective speed of the boat is reduced by the speed of the current, so we have:
84 km = (b - c) * 6 hours [Equation 1]
When traveling downstream (with the current), the effective speed of the boat is increased by the speed of the current, so we have:
84 km = (b + c) * 4.2 hours [Equation 2]
However, we need to convert the time of 4 hours and 12 minutes to decimal hours. 12 minutes is equal to 12/60 = 0.2 hours.
Substituting this value into Equation 2, we have:
84 km = (b + c) * 4.2 hours
Now, let's solve this system of equations to find the values of "b" and "c".
From Equation 1, we can rearrange it to solve for "c":
(b - c) * 6 = 84
b - c = 14 [Equation 3]
From Equation 2, we can rearrange it to solve for "c":
(b + c) * 4.2 = 84
b + c = 20 [Equation 4]
Now we can solve the system of equations 3 and 4 to find the values of "b" and "c".
Adding equations 3 and 4:
(b - c) + (b + c) = 14 + 20
2b = 34
Divide both sides by 2:
b = 17
Substituting the value of b into either Equation 3 or Equation 4, we can solve for "c":
17 + c = 20
c = 3
Therefore, the speed of the boat in still water is 17 km/h, and the speed of the current is 3 km/h.
To find the speed of the boat in still water (let's call it B) and the speed of the current (let's call it C), we can use the formula:
Distance = Speed × Time
Let's break down the problem into two parts:
1. The boat traveling up the river against the current:
- The speed of the boat relative to the water is B - C (since it's traveling against the current).
- The time it took to travel 84 km is 6 hours.
So, the equation for this part is: 84 = (B - C) × 6.
2. The boat traveling down the river with the current:
- The speed of the boat relative to the water is B + C (since it's traveling with the current).
- The time it took to travel 84 km is 4 hours and 12 minutes, which can be converted to 4 + 12/60 = 4.2 hours.
So, the equation for this part is: 84 = (B + C) × 4.2.
Now, we have two equations:
1. 84 = (B - C) × 6
2. 84 = (B + C) × 4.2
We can solve these equations simultaneously to find the values of B and C.
Let's start by simplifying the equations:
1. Simplifying equation 1: 84 = 6B - 6C
2. Simplifying equation 2: 84 = 4.2B + 4.2C
Now, we have a system of linear equations:
1. 6B - 6C = 84
2. 4.2B + 4.2C = 84
We can solve this system by using any method, such as substitution or elimination.
Let's solve by elimination:
Multiply equation 1 by 7 and equation 2 by 5:
1. 42B - 42C = 588
2. 21B + 21C = 420
Add the two equations:
42B - 42C + 21B + 21C = 588 + 420
63B = 1008
Divide both sides by 63:
B = 1008/63
B = 16
Now, substitute the value of B into equation 1 to solve for C:
6(16) - 6C = 84
96 - 6C = 84
-6C = 84 - 96
-6C = -12
Divide by -6:
C = -12/-6
C = 2
Therefore, the speed of the boat in still water (B) is 16 km/h, and the speed of the current (C) is 2 km/h.