Answered

Three towns P, Q and R are such that the distance between Q and R is double that between P and Q. A car covers the distance between P and Q with a speed 36 km/h and the distance between Q and R with a speed 48 km/h. The average speed for the entire journey is:​



Answer :

jacob193

Answer:

Approximately [tex]43.2[/tex] kilometers per hour.

Explanation:

To find average speed, divide the total distance travelled by the total duration of the motion:

[tex]\displaystyle (\text{average speed}) = \frac{(\text{total distance})}{(\text{total duration})}[/tex].

In this question, while the exact distance between the towns is not given, the ratio between the two distances is known. Let [tex]s[/tex] denote the distance (in kilometers, where [tex]s > 0[/tex]) between [tex]\sf P[/tex] and [tex]\sf Q[/tex]:

  • The distance between [tex]\sf Q[/tex] and [tex]\sf R[/tex] would be [tex]2\, s[/tex] (double the distance between [tex]\sf P[/tex] and [tex]\sf Q[/tex].)
  • The total distance travelled would be [tex]s + 2\, s = 3\, s[/tex].

To find the average speed, the total duration of the journey also needs to be found. To find total duration, start by finding the time required to cover each part of the journey:

  • Time required to cover a distance of [tex]s[/tex] at a speed of [tex]36[/tex] kilometers per hour: [tex](s / 36)[/tex] hours (between [tex]\sf P[/tex] and [tex]\sf Q[/tex].)
  • Time required to cover a distance of [tex]2\, s[/tex] at a speed of [tex]48[/tex] kilometers per hour: [tex](s / 48)[/tex] hours (between [tex]\sf Q[/tex] and [tex]\sf R[/tex].)

Hence, the total duration of the motion would be [tex]((s / 36) + (s / 48))[/tex] hours.

Divide the total distance travelled by the total duration to find average speed:

[tex]\begin{aligned} & (\text{average speed}) \\ =\; & \frac{(\text{total distance})}{(\text{total duration})} \\ =\; & \frac{s + 2\, s}{\displaystyle \frac{s}{36} + \frac{2\, s}{48}} \\ =\; & \frac{1 + 2}{\displaystyle \frac{1}{36} + \frac{2}{48}} && (\text{$s$ is eliminated}) \\ =\; & \frac{216}{5}\end{aligned}[/tex].

In other words, the average velocity of this journey would be [tex](216 / 5)[/tex] kilometers per hour, which is approximately [tex]43.2[/tex] kilometers per hour.