Question

A cyclist travels at $20$ kilometers per hour when cycling uphill, $24$ kilometers per hour when cycling on flat ground, and $30$ kilometers per hour when cycling downhill. On a sunny day, they cycle the hilly road from Aopslandia to Beast Island before turning around and cycling back to Aopslandia. What was their average speed during the entire round trip?

Answer 1

Answer:

Average speed during the trip = 24 km/h

Step-by-step explanation:

Given:

Speed of cyclist uphill, $v_1$ = 20 km/hr

Speed of cyclist on flat ground = 24 km/h

Speed of cyclist downhill, $v_2$ = 30 km/h

Cyclist has traveled on the hilly road to Beast Island from Aopslandia and then back to Aopslandia.

That means, one side the cyclist went uphill will the speed of 20 km/h and then came downhill with the speed of 30 km/h

To find:

Average speed during the entire trip = ?

Solution:

Let the distance between Beast Island and Aopslandia = D km

Let the time taken to reach Beast Island from Aopslandia = $T_1\ hours$

Formula for speed is given as:

$Speed = \dfrac{Distance}{Time}$

$v_1 = 20 = \dfrac{D}{T_1}$

$\Rightarrow T_1 = \dfrac{D}{20} ..... (1)$

Let the time taken to reach Aopslandia back from Beast Island = $T_2\ hours$

Formula for speed is given as:

$Speed = \dfrac{Distance}{Time}$

$v_2 = 30 = \dfrac{D}{T_2}$

$\Rightarrow T_2 = \dfrac{D}{30} ..... (2)$

Formula for average speed is given as:

$\text{Average Speed} = \dfrac{\text{Total Distance}}{\text{Total Time Taken}}$

Here total distance = D + D = 2D km

Total Time is $T_1+T_2$ hours.

Putting the values in the formula and using equations (1) and (2):

$\text{Average Speed} = \dfrac{2D}{T_1+T_2}}\\\Rightarrow \text{Average Speed} = \dfrac{2D}{\dfrac{D}{20}+\dfrac{D}{30}}}\\\Rightarrow \text{Average Speed} = \dfrac{2D}{\dfrac{30D+20D}{20\times 30}}\\\Rightarrow \text{Average Speed} = \dfrac{2D\times 20 \times 30}{{30D+20D}}\\\Rightarrow \text{Average Speed} = \dfrac{1200}{{50}}\\\Rightarrow \bold{\text{Average Speed} = 24\ km/hr}$

So, Average speed during the trip = 24 km/h