Q. 4.43 - Intermediate Algebra

Question

Translate to a system of equations and then solve:

A Mississippi river boat cruise sailed $120$ miles upstream for $12$ hours and then took $10$ hours to return to the dock.

Find the speed of the river boat in still water and the speed of the river current.

Step-by-Step Solution

Answer

The speed of the river boat in still water is $11$mph and the speed of the river current is $1$mph.

Step 1. Given

A river boat sailed $120$ miles upstream for $12$ hours and took $10$ hours to return to the dock.

Step 2. Translating into system of equations.

We know that distance is the product of rate and time.

So we will calculate the distance for the boat by assuming the speed or boat and river current as $b\&c$.

1.  When travelling upstream, the time taken is $12$ hours and the rate will be $(s-c)$. So the distance comes out to be $12(b-c)$.
2. When travelling downstream, the time taken is$10$ hours and the rate will be$(b+c)$. So the distance comes out to be $10(b+c)$.

We know the distance is $120$ miles.

Step 3. Finding the speed of the river boat in still water.

We have two equations, i.e.,
$$\begin{gathered} 10(b+c)=120 \\ 12(b-c)=120 \end{gathered}$$

Taking the common factors out give us the equations as-
$$\begin{gathered} b+c=12 \\ b-c=10 \end{gathered}$$

Adding both the equations give us $$\begin{gathered} 2b=22 \\ b=11 \end{gathered}$$.

This shows that the speed of boat in still water is $11 mph$.

Step 4. Finding the speed of river current.

We know that the speed of river boat in still water, i.e., $b=11$.

Putting this value in $b-c=10$ gives us,
$$\begin{gathered} 11-c=10 \\ c=1 \end{gathered}$$

This shows that the speed of water current is $1 mph$.

Step 5. Check the solution.

We will calculate the downstream and upstream rate by adding or subtracting the river current and the speed of boat and then calculate the distance covered. If the equation gives us the given value, our answers are right.

1. Downstream rate will become $11+1=12$ mph.
   In $10$ hours, the ship will have covered $10\times 12=120$ miles.
2. Upstream rate will become $11-1=10$ mph.
   In $12$ hours, the ship will have covered $12\times 10=120$ miles.

Thus, we get a given distance which shows our answer is right.