A van travels for 3 hours at an average speed of 40 mph. How much longer must the van travel at an average of 55 mph so that the average speed for the entire trip will be 50 mph?

6 hours

Can you explain how you got that please?

d1 = V1*T1 = 40 * 3 = 120 mi.

d2 = V2*T2 = 55 * T2 = 55T2.

Vavg. = (d1+d2)/(T1+T2) = 50 mi/h.
(120+55T2)/(3+T2) = 50.
120 + 55T2 = 150 + 50T2.
5T2 = 30.
T2 = 6 Hours.

To find out how much longer the van must travel, we first need to calculate the distance it has already traveled in the 3 hours at 40 mph.

Distance = Speed x Time
Distance = (40 mph) x (3 hours)
Distance = 120 miles

Now we can calculate the total distance the van needs to travel for the entire trip at an average speed of 50 mph. Let's represent the unknown distance as "D".

Total Distance = Distance at 40 mph + Distance at 55 mph
D = 120 miles + Distance at 55 mph

We know that the total trip time is 3 hours at 40 mph plus the extra time at 55 mph. So the total trip time can be expressed as:

Total Time = 3 hours + Extra Time

Since we want the average speed for the entire trip to be 50 mph, we can use the formula:

Average Speed = Total Distance / Total Time

Plugging in the known values:

50 mph = D / (3 hours + Extra Time)

To solve for "D", we can rearrange the formula:

D = 50 mph x (3 hours + Extra Time)

Now we can substitute this value of "D" back into the equation for the total distance:

50 mph x (3 hours + Extra Time) = 120 miles + Distance at 55 mph

Expand the equation:

150 mph + 50 mph x Extra Time = 120 miles + Distance at 55 mph

Subtract 120 miles from both sides:

150 mph + 50 mph x Extra Time - 120 miles = Distance at 55 mph

Combine the terms:

30 miles + 50 mph x Extra Time = Distance at 55 mph

Now we can use the formula for distance to find the remaining distance at 55 mph:

Distance = Speed x Time

Distance at 55 mph = (55 mph) x Extra Time

Substitute this into the equation:

30 miles + 50 mph x Extra Time = (55 mph) x Extra Time

Now we can solve for the Extra Time:

30 miles = (55 mph - 50 mph) x Extra Time
30 miles = 5 mph x Extra Time
Extra Time = 30 miles / 5 mph
Extra Time = 6 hours

Therefore, the van must travel for an extra 6 hours at an average speed of 55 mph in order to have an average speed of 50 mph for the entire trip.