Q. 360 - Intermediate Algebra

Question

In the following exercises,translate to a system of equations and solve.

Marcus can drive his boat 36 miles down the rever in  three hours to return upstream .Find the rate of the boat in still water and the rate of the current.

Step-by-Step Solution

Answer

The rate of the boat is still water is 10.5 mph and the rate of the current is 1.5 mph.

Step 1.Given information

Let x represents the rate of the boat in still water.

Let y represents the rate of the current.

The following chat will help us organize the data.

The boat goes downwards and the upstreams.Going downstream, the current helps the boat so the  boat's actual rate is $x+y$

Going upstream,the current slows the boat and so the actual rate is $x-y$.

Downstream it takes 3 hours.

upwords it takes 4 hours.

Each way the distance is 36 miles.

Name	Rate	Time	Distance D=r.t
Downstream	$x+y$	$3$	$$\begin{gathered} 3(x+y)=36 \\ (x+y)=12 \end{gathered}$$
Upstream	$x-y$	4	$$\begin{gathered} 4(x-y)=36 \\ (x-y)=9 \end{gathered}$$

Step 2.Solve for x.

From the equations 

$$\begin{gathered} x+y=12 \\ x-y=9 \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ 2x=21 \\ x=\frac{21}{2} \\ x=10.5 \end{gathered}$$

Step 3.Solve for y.

Substitute $x=10.5$ in the equation $x+y=12$.

$$\begin{gathered} x+y=12 \\ 10.5+y=12 \\ y=12-10.5 \\ y=1.5 \end{gathered}$$

Step 4.Check

Substitute $x=10.5,y=1.5$ in the equation $x-y=9.$

$$\begin{gathered} x-y=9 \\ 10.5-1.5=9 \\ 9=9 \\ this is true. \end{gathered}$$

Substitute $x=10.5,y=1.5$ into the equation $x+y=12.$

$$\begin{gathered} x+y=12 \\ 10.5+1.5=12 \\ 12=12 \\ This is true. \end{gathered}$$

Therefore,The rate of the boat is still water is 10.5 mph and the rate of the current is 1.5 mph.