A delivery van covers 90 km at 45 km/h, then 120 km at 60 km/h, and finally 150 km at z km/h. If its average speed for the entire trip is 60 km/h, determine z.

The correct option is 75 km/h.

Step-by-Step Explanation:

To find the value of speed $z$, we use the relation between total distance, total time, and average speed.

1. Calculate the time taken for each part of the journey:
Recall that $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$.

For the first part of the trip:

$t_1=\frac{90\text{km}}{45\text{km/h}}=2\text{hours}$

For the second part of the trip:

$t_2=\frac{120\text{km}}{60\text{km/h}}=2\text{hours}$

For the third part of the trip:

$t_3=\frac{150}{z}\text{hours}$

2. Calculate the total distance covered:

$\text{Total Distance}=90+120+150=360\text{km}$

3. Set up the equation for average speed:
The formula for average speed is:

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

We are given that the average speed for the entire trip is 60 km/h. Substituting the values into the formula gives:

$60=\frac{360}{2+2+\frac{150}{z}}$

$60=\frac{360}{4+\frac{150}{z}}$

4. Solve for $z$:
Rearranging the equation to solve for total time:

$4+\frac{150}{z}=\frac{360}{60}$

$4+\frac{150}{z}=6$

Subtract 4 from both sides:

$\frac{150}{z}=6-4$

$\frac{150}{z}=2$

Multiply both sides by $z$ and divide by 2:

$z=\frac{150}{2}=75\text{km/h}$

Thus, the required speed $z$ for the final segment is 75 km/h.