Round-Trip Speed Word Problem: Sarah's Boat & River Current

In summary, the question asks for the speed of the current in a river given the speed of a boat in still water and the time it takes to travel a certain distance. Using the relation distance equals speed times time, we can set up equations for the upstream and downstream portions of the trip and solve for the speed of the current. The final answer is that the current has a speed of 5 km/hr.
  • #1
MarkFL
Gold Member
MHB
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Here is the question:

Round-Trip Speed Word Problem help?

I'm having trouble and been stuck for half an hour just trying and erasing and I need some help.

Question: In still water, Sarah's boat can travel 15 km/h. If it takes her a total of 4 1/2 hours to travel 30 km up a river and then to return by the same route, what is the speed of the current in the river?

I have posted a link there to this thread so the OP can view my work.
 
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  • #2
Hello Meh,

Let's break this down into the upstream portion of the trip and the downstream portion, using the relation distance equals speed times time. We will let $0<c$ be the speed of the current and $t_1$ be the time going upstream and $t_2$ be the time going downstream.

\(\displaystyle 30=(15-c)t_1\)

\(\displaystyle 30=(15+c)t_2\)

Now, we also know the total time is 4.5 hours, so we may state:

\(\displaystyle t_1+t_2=4.5\implies t_2=4.5-t_1\)

And so our second equation becomes:

\(\displaystyle 30=(15+c)\left(4.5-t_1 \right)\)

Solving the first equation for $t_1$, we find:

\(\displaystyle t_1=\frac{30}{15-c}\)

Now, substituting this into our new second equation, we obtain an equation in one variable $c$, which is what we are trying to find:

\(\displaystyle 30=(15+c)\left(4.5-\frac{30}{15-c} \right)\)

Multiplying through by $15-c$, we get:

\(\displaystyle 30(15-c)=(15+c)\left(4.5(15-c)-30 \right)\)

Distribute on the right:

\(\displaystyle 30(15-c)=(15+c)\left(67.5-4.5c-30 \right)\)

Combine like terms:

\(\displaystyle 30(15-c)=(15+c)\left(37.5-4.5c \right)\)

Multiply through by $2$:

\(\displaystyle 30(30-2c)=(15+c)\left(75-9c \right)\)

Distribute on both sides:

\(\displaystyle 900-60c=1125-60c-9c^2\)

Combine like terms and arrange as:

\(\displaystyle 9c^2=225\)

Divide through by $9$:

\(\displaystyle c^2=25\)

Take the positive root:

\(\displaystyle c=5\)

Thus, we conclude that the current in the river is \(\displaystyle 5\,\frac{\text{km}}{\text{hr}}\).
 
  • #3

I'll solve this with one variable.


In still water, Sarah's boat can travel 15 km/hr.
If it takes her a total of 4 1/2 hours to travel 30 km
up a river and then to return by the same route,
what is the speed of the current in the river?

I will use: .[tex]\text{Time} \;=\;\frac{\text{Distance}}{\text{Speed}}[/tex]Let [tex]x[/tex] = speed of the current.

Going upstream, her speed is [tex]15-x[/tex] km/hr.
To travel 30 km, it takes: .[tex]\tfrac{30}{15-x}[/tex] hours.

Going downstream, her speed is [tex]15+x[/tex] km/hr.
To travel 30 km, it takes: .[tex]\tfrac{30}{15+x}[/tex] hours.

Her total time is [tex]4\!\tfrac{1}{2}[/tex] hours.

. . [tex]\frac{30}{15-x} + \frac{30}{15+x} \:=\:\frac{9}{2}[/tex]Multiply by [tex]2(15-x)(15+x)\!:[/tex]

. . [tex]\begin{array}{c}60(15+x) + 60(15-x) \:=\:9(15-x)(15+x) \\ \\ 900 + 60x + 900 - 60x \:=\:9(225-x^2) \\ \\ 1800 \:=\:2025 - 9x^2 \\ \\ 9x^2 \:=\:225 \\ \\ x^2 \:=\:25 \\ \\ x \:=\:5 \end{array}[/tex]The speed of the current is 5 km/hr.
 

FAQ: Round-Trip Speed Word Problem: Sarah's Boat & River Current

1. What is a round-trip speed word problem?

A round-trip speed word problem is a type of mathematical problem that involves calculating the speed of an object or person traveling both to and from a destination. It typically includes variables such as distance, time, and speed.

2. How does Sarah's boat and river current factor into this problem?

In this specific problem, Sarah's boat and the river current are two key variables that affect her overall speed. The river current is an external force that can either help or hinder her progress, while her boat's speed is a constant factor that remains the same throughout the entire trip.

3. What information do I need to solve this problem?

To solve this problem, you will need to know the distance of the trip, the speed of Sarah's boat, and the speed of the river current. You may also need to convert units of measurement if they are different (e.g. miles per hour to feet per second).

4. How do I set up the equation for this problem?

The equation for this problem is: total distance = (speed of boat + speed of current) * total time. You will need to plug in the known values for each variable and use algebra to solve for the unknown variable (either the speed of the boat or the speed of the current).

5. Are there any common mistakes to avoid when solving this type of problem?

Yes, some common mistakes to avoid include not properly converting units of measurement, using the wrong formula or equation, and not accounting for the different speeds of the boat and current during the round-trip. It is also important to carefully read and understand the problem before attempting to solve it.

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