An automated delivery drone flies from Logistics Center Alpha to Hub Beta at a uniform speed of 60 km/h and returns along the exact same route from Hub Beta back to Logistics Center Alpha at a speed of 40 km/h. What is the average speed of the drone for the entire round trip?

The correct answer is 48 km/h.

Step-by-step explanation:

To calculate the average speed of the drone for the complete round trip, we divide the total distance traveled by the total time taken.

Let the distance between Logistics Center Alpha and Hub Beta be d kilometers.

Step 1: Calculate the time taken for the forward journey
The drone flies from Logistics Center Alpha to Hub Beta at a uniform speed of 60 km/h.

$t_1=\frac{d}{60}\text{hours}$

Step 2: Calculate the time taken for the return journey
The drone returns along the exact same route from Hub Beta to Logistics Center Alpha at a speed of 40 km/h.

$t_2=\frac{d}{40}\text{hours}$

Step 3: Calculate the total distance and total time for the round trip
The total distance for the round trip is:

$\text{Total Distance}=d+d=2d\text{km}$

The total time for the round trip is:

$\text{Total Time}=t_1+t_2=\frac{d}{60}+\frac{d}{40}$

Taking a common denominator of 120 for the fractions:

$\text{Total Time}=\frac{2d+3d}{120}=\frac{5d}{120}=\frac{d}{24}\text{hours}$

Step 4: Calculate the average speed
Using the formula for average speed:

$\text{Average Speed}=\frac{\text{Total Distance}}{\text{Total Time}}$

Substitute the total distance and total time:

$\text{Average Speed}=\frac{2d}{\left(\frac{d}{24}\right)}$

$\text{Average Speed}=2\times 24=48\text{km/h}$

Alternative Method (Harmonic Mean Formula):
When equal distances are traveled at two different speeds v₁ and v₂, the average speed is given by their harmonic mean:

$v_{\text{avg}}=\frac{2v_1{v}_2}{v_1+v_2}$

Substituting v₁ = 60 km/h and v₂ = 40 km/h:

$v_{\text{avg}}=\frac{2\times 60\times 40}{60+40}=\frac{4800}{100}=48\text{km/h}$

Thus, the average speed of the drone for the entire round trip is 48 km/h.