Q. 59 - Precalculus Enhanced with Graphing Utilities

Question

On a recent trip to the Cuyabeno Wildlife Reserve in the Amazon region of Ecuador, Mike took a $100$-kilometer trip by speedboat down the Aguarico River from Chiritza to the Flotel Orellana. As Mike watched the Amazon unfold, he wondered how fast the speedboat was going and how fast the current of the white water Aguarico River was. Mike timed the trip downstream at $2.5$ hours and the return trip at $3$ hours. What were the two speeds?

Step-by-Step Solution

Answer

The speed of the speed boat is $36.67 km/hr$ and the speed of the river current is $3.33 km/hr$

Step 1. Given

Total distance by speed boat is $100 km$

Mike timed the trip downstream at $2.5$ hours and the return trip at $3$ hours.

Step 2. Find the speed for each trip

We know that,

Speed$=\frac{Dis\tan ce covered}{Time taken}$

Downstream speed$$\begin{gathered} =\frac{100}{2.5} \\ =40 \end{gathered}$$

Upstream speed$=\frac{100}{3}$

                           $\simeq 33.33$

Step 3. Form the system of equations.

Generally the downstream speed is the sum of the speed of the boat and the current of the river.

Similarly, the upstream speed is the difference of the speed of the boat and the current of the river. 

If $x,y$ denotes the speed of the boat and the current of the river respectively, then,

$\left\{\begin{array}{l}x+y=40\\ x-y=33.33\end{array}\right.$

Step 4. Solve the system

Solve the system of equations.

Add both the equations,

$$\begin{gathered} 2x=73.33 \\ x=36.665 \end{gathered}$$

Step 5. Find y

Substitute $x=36.665$ in the first equation,

$$\begin{gathered} x+y=40 \\ 36.67+y=40 \\ y=40-36.67 \\ =3.33 \end{gathered}$$

Thus the speed of the boat is $36.67 km/hr$ and the river current is $3.33 km/hr$