A courier van covers 90 km at 30 km/h, then 100 km at 50 km/h, and finally 30 km at v km/h. If its average speed for the entire trip is 40 km/h, determine v.

The correct option is 60 km/h.

Step 1: Understand the concept of Average Speed
The average speed for a trip consisting of multiple legs is calculated by dividing the total distance covered by the total time taken for the entire trip.

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

Step 2: Calculate the time taken for each part of the journey
Time taken for any segment is given by the formula:

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

Let us break down the courier van's trip into three parts:

Part 1:
Distance, $d_1=90\text{km}$
Speed, $v_1=30\text{km/h}$
Time taken, $t_1=\frac{90}{30}=3\text{hours}$

Part 2:
Distance, $d_2=100\text{km}$
Speed, $v_2=50\text{km/h}$
Time taken, $t_2=\frac{100}{50}=2\text{hours}$

Part 3:
Distance, $d_3=30\text{km}$
Speed, $v_3=v\text{km/h}$
Time taken, $t_3=\frac{30}{v}\text{hours}$

Step 3: Calculate the Total Distance and Total Time
Total Distance ($D$):

$D=90+100+30=220\text{km}$

Total Time ($T$):

$T=t_1+t_2+t_3=3+2+\frac{30}{v}=5+\frac{30}{v}\text{hours}$

Step 4: Set up and solve the equation for v
Given that the average speed for the entire trip is 40 km/h, substitute the values into the average speed formula:

$40=\frac{220}{5+\frac{30}{v}}$

Rearrange the terms to solve for the denominator:

$5+\frac{30}{v}=\frac{220}{40}$

Simplify the fraction on the right-hand side:

$5+\frac{30}{v}=5.5$

Subtract 5 from both sides:

$\frac{30}{v}=5.5-5$

$\frac{30}{v}=0.5$

Solve for $v$:

$v=\frac{30}{0.5}=60\text{km/h}$

Therefore, the speed v is equal to 60 km/h.