It takes Ashley 8 hours to kayak 18 km up river and 18 km back down the river. if the river is flowing at 3 km/h, how fast did she puddle assuming she paddled at a constant speed?
her speed = v
current speed = 3
time up = 18/(v-3)
time down = 18/(v+3)
so
18[ 1/(v-3) + 1/(v+3) ] = 8
[ 1/(v-3) + 1/(v+3) ] = 4/9
[ v + 3 + v - 3 ] = (4/9)(v^2-9)
18 v = 4 v^2 - 36
2 v^2 - 9 v - 18 = 0
v = 6 for positive v root
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check
up speed = 3
down speed = 9
up time = 18/3 = 6
down time = 18/9 = 2
total = 8 hr ok
Well, Ashley must be quite skilled to kayak upriver and back down without getting all tangled up in the water! If the river is flowing at 3 km/h and she completes a total distance of 36 km (18 km up + 18 km down), then we need to subtract the effect of the river's flow. Since Ashley takes 8 hours for the round trip, we can calculate her paddling speed by dividing the total distance by the total time. So, without further ado, the math:
Total distance: 36 km
River's flow: 3 km/h
Total time: 8 hours
To account for the river's flow, we subtract its speed from the total speed:
Total speed = Paddling speed - River's flow
Now, let's find the paddling speed:
Total speed = 36 km / 8 hours
Total speed = 4.5 km/h
Putting it all together:
4.5 km/h = Paddling speed - 3 km/h
So, after some hilarious math, it appears that Ashley's paddling speed was approximately 7.5 km/h! Now that's some impressive puddling!
To find out how fast Ashley paddled, we need to calculate the time it took her to kayak against the current (upstream) and with the current (downstream). We can then determine her paddling speed by subtracting the speed of the river from the average speed of the kayak.
Let's break down the problem step-by-step:
1. Calculate the time it took Ashley to kayak upstream:
- Distance: 18 km
- Speed of the river: 3 km/h
- Relative speed of the kayak against the current: Speed of the kayak - Speed of the river
- Time = Distance / Relative speed
- Time = 18 km / (Speed of the kayak - 3 km/h)
2. Calculate the time it took Ashley to kayak downstream:
- Distance: 18 km
- Speed of the river: 3 km/h
- Relative speed of the kayak with the current: Speed of the kayak + Speed of the river
- Time = Distance / Relative speed
- Time = 18 km / (Speed of the kayak + 3 km/h)
3. Total time spent kayaking: 8 hours
- Total time = Time upstream + Time downstream
- 8 hours = [18 km / (Speed of the kayak - 3 km/h)] + [18 km / (Speed of the kayak + 3 km/h)]
4. Solve the equation to find the speed of the kayak:
- Multiply both sides of the equation by (Speed of the kayak - 3 km/h) * (Speed of the kayak + 3 km/h)
- 8 hours * (Speed of the kayak - 3 km/h) * (Speed of the kayak + 3 km/h) = 18 km * (Speed of the kayak + 3 km/h) + 18 km * (Speed of the kayak - 3 km/h)
5. Simplify the equation:
- 8(Speed of the kayak^2 - 9 km^2) = 36(Speed of the kayak) + (-54) km
- 8Speed of the kayak^2 - 72 km^2 = 36Speed of the kayak - 54 km
6. Rearrange the equation to solve for Speed of the kayak:
- 8Speed of the kayak^2 - 36Speed of the kayak - 72 km^2 + 54 km = 0
7. Solve the quadratic equation using factoring, completing the square, or the quadratic formula to find the two possible speeds of the kayak.
By solving this quadratic equation, we can find the possible speeds at which Ashley paddled.
To determine the speed at which Ashley paddled, we need to subtract the velocity of the river from the given values. Let's break down the problem into two parts: Ashley kayaking upstream and Ashley kayaking downstream.
1. Upstream:
When Ashley kayaks upstream, the speed of the river needs to be subtracted from her actual speed, as it opposes her motion. Let's denote Ashley's paddling speed as "P". So, her effective speed while kayaking upstream will be P - 3 km/h (since the river flows at 3 km/h in the opposite direction).
The distance covered while kayaking upstream is 18 km.
Using the formula Speed = Distance / Time, we can create an equation for Ashley's upstream journey:
(P - 3) km/h = 18 km / (Time taken upstream)
2. Downstream:
When Ashley kayaks downstream, the speed of the river needs to be added to her actual speed, as it assists her motion. So, her effective speed while kayaking downstream will be P + 3 km/h.
The distance covered while kayaking downstream is also 18 km.
Using the same formula, let's create an equation for Ashley's downstream journey:
(P + 3) km/h = 18 km / (Time taken downstream)
Since the total time for the complete journey is given as 8 hours, we can write:
(Time taken upstream) + (Time taken downstream) = 8 hours
Now, we have three equations:
(P - 3) km/h = 18 km / (Time taken upstream)
(P + 3) km/h = 18 km / (Time taken downstream)
(Time taken upstream) + (Time taken downstream) = 8 hours
Solving these three equations will give us the value of P, which is the speed at which Ashley paddled.