The point of no return for an airplane, flying over water from point A on land to point B on land, is that distance into the trip for which it takes just as much time to go on to B as it does to return to A. The distance from San Francisco to Honolulu is 2387 miles. A plane leaves San Francisco at a speed, in still air, of 400 mi/h. There is a 60 mi/h tail wind.

a. Find the point of no return.

b. After traveling 1042 miles, the pilot of the plane determines that it is necessary to make an emergency landing. Would it require less time to continue to Honolulu or to return to San Francisco?

I assume the point of no return over water is the same as over land :-)

At x miles out, the time to Honolulu is (2387-x)/(400+60) and the time to go back to SF is x/(400-60)

(2387-x)/460 = x/340
811580 - 340x = 460x
800x = 811580
x = 1014 miles from SF

at 1042 miles, it is obviously quicker to fly on to Honolulu, since it's past the PNR.

a. To find the point of no return, we need to calculate the time it takes to go from San Francisco to the point of no return and compare it with the time it takes to return from that point to San Francisco.

Let's first find the speed of the plane with the tailwind:
Speed of the plane with the tailwind = Speed in still air + Tailwind speed
= 400 mi/h + 60 mi/h
= 460 mi/h

Now, we can calculate the time it takes to travel from San Francisco to the point of no return:
Time = Distance / Speed
Time to point of no return = Distance to point of no return / Speed of the plane with tailwind
= x miles / 460 mi/h
= x/460 hours

Since the speed is the same going back to San Francisco, the time taken to return from the point of no return to San Francisco is also x/460 hours.

According to the problem statement, the point of no return is the distance into the trip for which it takes just as much time to go on to B as it does to return to A. Therefore, the time taken to travel from San Francisco to the point of no return is equal to the time taken to return from the point of no return to San Francisco.

Hence, x/460 = x/460

The distance from San Francisco to Honolulu is 2387 miles, so the point of no return would be at a distance of x miles from San Francisco.

Solving the equation x/460 = x/460, we find that x = 2387/2 = 1193.5 miles.

Therefore, the point of no return is located approximately 1193.5 miles from San Francisco.

b. After traveling 1042 miles, the pilot of the plane determines that it is necessary to make an emergency landing. To determine whether it would require less time to continue to Honolulu or to return to San Francisco, we need to compare the time it would take to reach either destination.

Time taken to reach Honolulu = (Total distance - Distance traveled) / Speed
= (2387 - 1042) miles / 460 mi/h
= 1345 miles / 460 mi/h
= 2.92 hours

Time taken to return to San Francisco from the point of emergency landing = Distance traveled / Speed
= 1042 miles / 460 mi/h
= 2.26 hours

Therefore, it would require less time to return to San Francisco from the emergency landing point rather than continuing to Honolulu.

a. To find the point of no return, we need to determine the distance at which it takes the same amount of time to go on to B as it does to return to A.

Let's denote the point of no return as x miles from San Francisco (point A). The distance from the point of no return to Honolulu (point B) would then be (2387 - x) miles.

The speed of the plane with the tailwind is the sum of the plane's airspeed and the wind speed. Given that the plane's airspeed is 400 mi/h and the tailwind is 60 mi/h, the actual speed of the plane relative to the ground is 400 + 60 = 460 mi/h.

The time it takes to complete the trip from San Francisco to the point of no return is equal to the distance divided by the speed:
Time 1 = x / 460.

The time it takes for the return trip from the point of no return to San Francisco is also equal to the distance divided by the speed, but since the plane is now flying against the headwind, the effective speed would be the airspeed minus the headwind speed:
Time 2 = (2387 - x) / (400 - 60) = (2387 - x) / 340.

To find the point of no return, we set Time 1 equal to Time 2:
x / 460 = (2387 - x) / 340.

Now we can solve this equation for x:
340x = 460(2387 - x)
340x = 1098140 - 460x
800x = 1098140
x = 1372.675.

Therefore, the point of no return is approximately 1372.675 miles from San Francisco.

b. To determine whether it would require less time to continue to Honolulu or to return to San Francisco after traveling 1042 miles, we need to compare the remaining distance to each destination.

After traveling 1042 miles from San Francisco, the remaining distance to Honolulu would be 2387 - 1042 = 1345 miles.

To calculate the time to continue to Honolulu, we can use the plane's airspeed of 400 mi/h:
Time to Honolulu = 1345 / 400 = 3.3625 hours.

To calculate the time to return to San Francisco, we need to consider the headwind. The effective speed of the plane while flying against the headwind is the airspeed minus the headwind speed, which is 400 - 60 = 340 mi/h:
Time to San Francisco = 1042 / 340 = 3.0647 hours.

Therefore, it would require less time to return to San Francisco after traveling 1042 miles.