Question

Two boats start together and race across a 42-km-wide lake and back. Boat A goes across at 42 km/h and returns at 42 km/h. Boat B goes across at 21 km/h, and its crew, realizing how far behind it is getting, returns at 63 km/h. Turnaround times are negligible, and the boat that completes the round trip first wins.(a) Which boat wins?o Boat Ao Boat Bo It's a tie(b) By how much?_____ km(c) What is the average velocity of the winning boat?_____ km/h

Answer 1

`\begin{gathered} (a)\text{ Boat A} \\ \\ (b)\text{ }42\text{ km} \\ \\ (c)\text{ }0\text{ km/h} \end{gathered}`

Step-by-step explanation

(a) To determine which boat wins, we have to find the total time the journey takes for each boat.

To do that, apply the formula for speed:

`\begin{gathered} s=(d)/(t) \\ \\ t=(d)/(s) \end{gathered}`

where d = distance traveled

t = time taken

For boat A, the time taken to go across and return is:

`\begin{gathered} t=(42)/(42)+(42)/(42)=1+1 \\ \\ t=2\text{ hrs} \end{gathered}`

For boat B, the time taken to go across and return is:

`\begin{gathered} t=(42)/(21)+(42)/(63)=2+0.667 \\ \\ t=2.667\text{ hrs} \end{gathered}`

We see that it takes boat B longer to complete the race. Hence, boat A will win the race.

(b) To determine how much boat A will win, we can look at the distances both boats cover. Notice that while getting across the lake, boat A travels twice as fast as boat B and travels at the same speed while returning.

This implies that in the time that boat A travels both laps of the race, boat B is just reaching the other end of the lake.

In other words, in the time that boat A travels 84 km, boat B travels 42 km.

Hence, boat A will beat boat B by a distance of:

`\begin{gathered} 84-42\text{ km} \\ \\ 42\text{ km} \end{gathered}`

(c) In the race, the winning boat starts from one end of the lake, gets to the other end, and then, returns to the starting point.

This implies that the displacement of the boat is 0 km.

Velocity is given by dividing displacement by the time taken. Since the displacement is 0 km, the velocity of the winning boat, Boat A is 0 km/h