Chapter 4.6, Problem 45E

To determine

To find: The speed of barge in water.

Answer to Problem 45E

The speed of the barge is $8\text{mph}$ .

Explanation of Solution

Given; The speed of the Puget in sound is $5\text{miles/hour}$ . A barge travels with the current $26$ miles and return in $10\frac{2}{3}\text{hours}$ .

The distance is given by,

  $d=rt$ .

Here,

The speed of the current is $5\text{}\text{miles/hour}$ .

With the current add the speed of the current to the rate and against the current subtracts the speed of the current from the rate.

Hence,

With current

  $\begin{array}{c}\left(r+5\right)t=26\\ t=\frac{26}{r+5}\end{array}$

Against the current

  $\begin{array}{c}\left(r-5\right)t=26\\ t=\frac{26}{r-5}\end{array}$

The total time is given by adding with current and against current.

Thus,

  $\frac{32}{3}=\frac{26}{r+5}+\frac{26}{r-5}$ .

On solving,

  $\begin{array}{c}26\left(3\right)\left(r-5\right)+26\left(3\right)\left(r+5\right)=32\left(r-5\right)\left(r+5\right)\\ 78r-390+78r+390=32\left(r^2-25\right)\\ 156r=32r^2-800\\ r=\frac{-25}{8},8\end{array}$

There are two values of $r$ but speed cannot be in negative.

Therefore, the speed of the barge is $8\text{mph}$ .