Question 6
Luiza kayaks from her house to visit her friend Fernando, who lives 16.2 miles downriver. On her way to Fernando’s house, traveling with the current, the trip takes her an hour and a half. The trip back home takes 13 and a half hours. What is the speed of the current, in miles per hour?
(Suppose that Luiza paddles at the same speed and the river moves at the same speed for both the trip to the park and the trip home.)
man how you gonna give the answer without giving us the answer we want
Not a very useful answer
I apologize. Here is the full solution:
Let's denote the speed of Luiza as $p$ and the speed of the current as $c$.
Using the formula $d = rt$ (distance equals rate times time), we can write two equations based on the two different trips:
$\text{downstream: } 16.2 = (p+c)\cdot \frac{3}{2}$
$\text{upstream: } 16.2 = (p-c)\cdot 13.5$
Simplifying the first equation, we get:
$16.2 = \frac{3}{2}p + \frac{3}{2}c$
Multiplying both sides by 2/3, we get:
$10.8 = p+c$
Simplifying the second equation, we get:
$16.2 = 13.5p - 13.5c$
Dividing both sides by 13.5, we get:
$1.2 = p - c$
Now we have two equations:
$10.8 = p+c$
$1.2 = p - c$
We can add these equations to eliminate $c$:
$12 = 2p$
$p= 6$
Substituting this value of $p$ into either equation gives us:
$10.8 = 6+c$
$c = 4.8$
Therefore, the speed of the current is 4.8 miles per hour.
Well, let's think about this. If Luiza is traveling with the current, it means the current is helping her go faster. That's like having a personal water slide, but without the water guns. On the return trip, she's going against the current, so it's like someone filled the river with molasses. Not quite as fun, eh?
Now, let's do some math. We know that the distance from Luiza's house to Fernando's is 16.2 miles. We also know that it took her an hour and a half on the way there and 13 and a half hours on the way back. So, we can use the formula: speed = distance / time.
On the way there, Luiza's speed (including the speed of the current) is 16.2 miles / 1.5 hours = 10.8 miles per hour.
On the way back, Luiza's speed (against the current) is 16.2 miles / 13.5 hours = 1.2 miles per hour.
Now, since we know that Luiza's paddling speed is the same for both trips, let's call it "x" miles per hour. And we'll call the speed of the current "c" miles per hour.
So, Luiza's speed on the way there is x + c and her speed on the way back is x - c.
Using the information we've gathered, we can set up two equations:
x + c = 10.8
x - c = 1.2
Now, let's solve this fun puzzle. We can add the two equations together:
(x + c) + (x - c) = 10.8 + 1.2
2x = 12
Dividing both sides by 2, we find:
x = 6
So, Luiza's paddling speed is 6 miles per hour, and her friend, the river, is flowing at a speed of 4.8 miles per hour with the current. That current sure knows how to have a good time!
To find the speed of the current, we can use the formula:
Speed = Distance / Time
Let's first find the speed of Luiza's kayak when traveling with the current (from her house to Fernando's house).
Distance = 16.2 miles
Time = 1.5 hours
Speed (with the current) = Distance / Time
Speed (with the current) = 16.2 miles / 1.5 hours
Speed (with the current) ≈ 10.8 miles per hour
Now, let's find the speed of Luiza's kayak when traveling against the current (from Fernando's house back to her house).
Distance = 16.2 miles
Time = 13.5 hours
Speed (against the current) = Distance / Time
Speed (against the current) = 16.2 miles / 13.5 hours
Speed (against the current) ≈ 1.2 miles per hour
Since the current affects the speed of the kayak, we can calculate the speed of the current by finding the difference between the speed of the kayak when traveling with the current and the speed of the kayak when traveling against the current.
Speed of current = Speed (with the current) - Speed (against the current)
Speed of current = 10.8 miles per hour - 1.2 miles per hour
Speed of current ≈ 9.6 miles per hour
Therefore, the speed of the current is approximately 9.6 miles per hour.
since time = distance/speed, if
she paddles at p mi/hr
and the current flows at c mi/hr
then
16.2/(p+c) = 3/2
16.2/(p-c) = 27/2
Now just solve for c