Jamie trains on his surfs every weekend. He paddles upstream for 3 1/2 hours, and then returns downstream to the point where he started in 2 hours and 6 minutes. If the river flows at 3 km/h find:
a) his paddling speed in still water
upstream speed = u - c
downstream speed = u + c
d = (u-c)(3.5) = (u+c)(2.1)
but c = 3
so
(u-3)(3.5)=(u+3)(2.1)
3.5 u - 10.5 = 2.1 u + 6.3
1.4 u = 16.8
a)paddling speed in still water
distance = 10.5 km upstream only
time =3.5 hours
s=d/t
10.5 divided by 3.5 = 3km/h
b)the distance he paddles upstream
distance=3.5 hours
To find Jamie's paddling speed in still water, we can use the concept of relative velocity.
Let's say Jamie's paddling speed in still water is \(v\) km/h, and the speed of the river is 3 km/h. When paddling upstream, Jamie's effective speed against the current is the difference of his paddling speed and the speed of the river, i.e., \(v - 3\) km/h.
Furthermore, when paddling downstream, Jamie's effective speed is the sum of his paddling speed and the speed of the river, i.e., \(v + 3\) km/h.
We know that Jamie spends 3 1/2 hours paddling upstream and 2 hours and 6 minutes returning downstream.
Converting 3 1/2 hours to minutes:
3 1/2 hours = 3 hours + 0.5 hours = 3*60 minutes + 0.5*60 minutes = 180 + 30 = 210 minutes.
Converting 2 hours and 6 minutes to minutes:
2 hours + 6 minutes = 2 * 60 minutes + 6 minutes = 120 + 6 = 126 minutes.
Upstream time = 210 minutes
Downstream time = 126 minutes
The distance Jamie covered while paddling upstream is the same as the distance covered when returning downstream.
Now, we can use the formula:
Distance = Speed * Time
The distance covered when paddling upstream is \((v - 3) \times 210\) km, and the distance covered when paddling downstream is \((v + 3) \times 126\) km.
Since the distances are the same, we can write the equation:
\((v - 3) \times 210 = (v + 3) \times 126\)
Simplifying this equation:
\(210v - 630 = 126v + 378\)
Let's solve for \(v\):
\(210v - 126v = 378 + 630\)
\(84v = 1008\)
Dividing both sides of the equation by 84:
\(v = \frac{1008}{84}\)
\(v = 12\)
Therefore, Jamie's paddling speed in still water is 12 km/h.
To find Jamie's paddling speed in still water, we can use the concept of relative velocity.
Let's denote Jamie's paddling speed in still water as "x" km/h.
When Jamie is paddling upstream, he has to fight against the current. The river is flowing at a speed of 3 km/h, so his effective speed against the current is (x - 3) km/h.
The distance covered by Jamie while paddling upstream is the product of his effective speed against the current and the time taken. Given that he paddles for 3 1/2 hours, which is 3.5 hours, the distance covered upstream is (3.5)(x - 3) km.
When Jamie paddles downstream, the current assists him. So his effective speed with the current is (x + 3) km/h.
The distance covered by Jamie while paddling downstream is the product of his effective speed with the current and the time taken. Given that he paddles for 2 hours and 6 minutes, which is 2.1 hours, the distance covered downstream is (2.1)(x + 3) km.
Since Jamie paddles back to the point where he started, the distance covered upstream and downstream should be equal. Therefore, we can set up the following equation:
(3.5)(x - 3) = (2.1)(x + 3)
Now we can solve this equation to find the value of x, which represents Jamie's paddling speed in still water.
Expanding both sides of the equation:
3.5x - 10.5 = 2.1x + 6.3
Combining like terms:
3.5x - 2.1x = 6.3 + 10.5
1.4x = 16.8
Dividing both sides of the equation by 1.4:
x = 16.8 / 1.4
x = 12
Therefore, Jamie's paddling speed in still water is 12 km/h.