How do I solve this?

A plane flying the 3458-mi from New York City to London has a 50-MPH tailwind. The flight's point of no return is the point at which the flight time required to return to New York is the same as the time required to continue to London. If the Plane's speed in still air is 850 mph, how far is New York from the point of no return?

If the point is x miles from NYC, then since time = distance/speed,

(3458-x)/(850+50) = x/(850-50)

To solve this problem, we can first find the time it takes to fly from New York to London with the tailwind, and then calculate the distance from New York to the point of no return.

Step 1: Find the time taken to fly from New York to London with the tailwind:
For this, we'll use the formula:
Time = Distance / Speed.

The distance from New York to London is 3458 miles. The speed of the plane with the tailwind is the sum of the plane's speed in still air (850 mph) and the tailwind speed (50 mph). Therefore, the speed of the plane with the tailwind is 850 mph + 50 mph = 900 mph.

Time taken to fly from New York to London with the tailwind:
Time = 3458 miles / 900 mph.

Step 2: Find the time taken to fly from London to New York against the headwind:
Since the plane's speed in still air is 850 mph, and we know the tailwind is 50 mph, the headwind will be 50 mph less than the speed in still air, which is 850 mph - 50 mph = 800 mph.

Time taken to fly from London to New York with the headwind:
Time = Distance / Speed.

Step 3: Find the distance from New York to the point of no return:
The point of no return is the point at which the time required to return to New York is the same as the time required to continue to London. This means that the time taken from London to New York with the headwind (found in Step 2) is the same as the time taken from New York to London with the tailwind (found in Step 1).

Now, let's calculate the time taken from London to New York with the headwind:

Time = Distance / Speed
Time = Distance / (850 mph - 50 mph)
Time = Distance / 800 mph

Since the time taken from London to New York is the same as the time taken from New York to London, we can set these two equations equal to each other:

Distance / 900 mph = Distance / 800 mph

To solve for the distance from New York to the point of no return, we can cross-multiply:

800 * Distance = 900 * Distance

800 * Distance - 900 * Distance = 0
(900 - 800) * Distance = 0
100 * Distance = 0

Distance = 0

Therefore, the distance from New York to the point of no return is 0 miles. This means that the point of no return is the starting point in New York.

To solve this problem, we need to determine the flight time required to return to New York and the flight time required to continue to London.

Let's start by calculating the time it takes to fly from New York to London with the current conditions. We can use the formula:

Time = Distance / Speed

The distance from New York to London is given as 3458 miles. Given that the plane's speed in still air is 850 mph and it has a 50 mph tailwind, the effective speed of the plane is (850 + 50) mph = 900 mph.

Plugging these values into the formula, we get:

Time from New York to London = 3458 miles / 900 mph ≈ 3.843 hours (rounded to three decimal places)

Next, we need to find the time required to return from London to New York with the same conditions. Since the plane is flying against the wind, we need to subtract the tailwind speed from the plane's speed in still air.

Effective speed when flying against the wind = Plane's speed in still air - Tailwind speed

Effective speed when flying against the wind = 850 mph - 50 mph = 800 mph

Using the formula, we can calculate:

Time from London to New York = 3458 miles / 800 mph ≈ 4.323 hours (rounded to three decimal places)

The point of no return is the point when the time required to return to New York is the same as the time required to continue to London. So, to find the distance from the point of no return to New York, we can subtract the time to return from the total distance:

Distance from the point of no return to New York = (Time from New York to London - Time from London to New York) * (Speed from New York to London)

Distance from the point of no return to New York = (3.843 hours - 4.323 hours) * (900 mph) ≈ -0.480 hours * (900 mph) ≈ -432 miles

The negative distance indicates that the point of no return is approximately 432 miles beyond London when flying from New York to London with this tailwind.

Therefore, the distance from New York to the point of no return is approximately 3458 miles - 432 miles ≈ 3026 miles.

d = f(p-t) / 2p

now let's see the numbers
d = 3458(850-50) / 2(850)
d = 3458(800) / 1700
d = 2766400 / 1700
d = approx. 1627.29