A plane is flying on a course with a vector of <250, 100>, where the 250 is the speed due north, and the 100 is the speed due east. The wind is blowing <25, 5> where the directions are also north and east, in the same order. How fast is the plane flying? All values are in mph.

final course vector

<275,105>

magnitude = √(275^2+105^2)
= 294.36 units of speed

To find the speed of the plane, we can use the concept of vector addition. The velocity of the plane is the sum of its speed due north and its speed due east. Similarly, the velocity of the wind is the sum of its speed due north and its speed due east.

Given the course vector of the plane <250, 100>, we can break it down into its north and east components.

North component of the plane's velocity = 250 mph
East component of the plane's velocity = 100 mph

Similarly, the wind vector <25, 5> can also be broken down into its north and east components.

North component of the wind velocity = 25 mph
East component of the wind velocity = 5 mph

To calculate the total speed of the plane, we need to add the north and east components of the plane's velocity, while taking into account the north and east components of the wind.

Total north component = North component of the plane's velocity + North component of the wind velocity
Total north component = 250 mph + 25 mph = 275 mph

Total east component = East component of the plane's velocity + East component of the wind velocity
Total east component = 100 mph + 5 mph = 105 mph

Now, we can use the Pythagorean theorem to calculate the speed of the plane.

Speed of the plane = √((Total north component)^2 + (Total east component)^2)
Speed of the plane = √((275 mph)^2 + (105 mph)^2)
Speed of the plane = √(75625 mph^2 + 11025 mph^2)
Speed of the plane = √(86650 mph^2)
Speed of the plane ≈ 294.5 mph

Therefore, the plane is flying at a speed of approximately 294.5 mph.