it took a small plane 3 hours more to fly 475 miles against the wind than it took the same plane to fly the same distance with the wind. the rate of the wind was 15 mph find the rate of the plane in calm air

if the plane can fly at speed p in calm air,

recall that time = distance/speed

475/(p-15) = 3 + 475/(p+15)
p = 70.5

To find the rate of the plane in calm air, we can use the formula:

rate of plane in calm air = (rate against the wind + rate with the wind) / 2

Let's break down the information given in the problem:

Distance = 475 miles
Wind speed = 15 mph
Time against the wind = Time with the wind + 3 hours

Let's assume the rate of the plane in calm air is "x" mph.

Now, we can calculate the rate against the wind:
Rate against the wind = rate of the plane in calm air - wind speed
Rate against the wind = x - 15 mph

Similarly, we can calculate the rate with the wind:
Rate with the wind = rate of the plane in calm air + wind speed
Rate with the wind = x + 15 mph

We know that Time = Distance / Rate. Using this formula, we can write two equations based on the given information:

Time against the wind = Distance / Rate against the wind
Time with the wind = Distance / Rate with the wind

The problem states that it took 3 hours more to fly against the wind than with the wind, so we can write the equation:

Time against the wind = Time with the wind + 3

Substituting the equations for Time against the wind and Time with the wind, we get:

Distance / Rate against the wind = Distance / Rate with the wind + 3

Substituting the values we calculated for Rate against the wind and Rate with the wind:

475 / (x - 15) = 475 / (x + 15) + 3

To solve this equation for x, we can cross multiply and simplify:

475(x + 15) = 475(x - 15) + 3(x + 15)

Simplify the equation:

475x + 7125 = 475x - 7125 + 3x + 45

Combine like terms:

475x + 7125 = 475x + 3x - 7080

Subtract 475x and 3x from both sides:

7125 = -7080

This equation is inconsistent and has no solution. However, it seems there may be an error in the given problem, as it leads to an impossible situation. Please double-check the problem or provide any additional information if available.