A plane is headed due south with an airspeed of 190 mph. A wind from the direction of S 40 degrees W is blowing at 30 mph. Find the groundspeed of the plane, rounded to the nearest whole number

s = speed south = 190 - 30 cos 40

w = speed west = 30 sin 40

speed magnitude = sqrt (s^2 + w^2)

To find the groundspeed of the plane, we need to consider the effect of the wind on the plane's motion.

First, let's break down the motion of the plane into its components.
The plane is moving due south, so its direction is 180 degrees (opposite to north). The airspeed of the plane is 190 mph in this direction.

The wind is blowing from the direction of S 40 degrees W (south 40 degrees west), which means it is coming from the direction of 40 degrees east of south. Therefore, the wind's direction is 180 + 40 = 220 degrees.

To calculate the effect of the wind on the plane's motion, we can use vector addition. We add the vector representing the airspeed of the plane and the vector representing the wind speed.

Using the law of cosines, we can calculate the magnitude of the resultant vector (the groundspeed of the plane) as follows:

groundspeed^2 = airspeed^2 + wind^2 - 2 * airspeed * wind * cos(angle between them)

where airspeed = 190 mph and wind = 30 mph.

angle between the airspeed and wind vectors = angle between the direction of the plane and the wind's direction = 180 degrees - 220 degrees = -40 degrees (negative because it is measured counterclockwise).

Substituting these values into the equation:

groundspeed^2 = (190)^2 + (30)^2 - 2 * 190 * 30 * cos(-40 degrees)

Simplifying:

groundspeed^2 = 36100 + 900 - 2 * 190 * 30 * cos(-40 degrees)

Using cosine function (cos(-x) = cos(x)):

groundspeed^2 = 36100 + 900 - 2 * 190 * 30 * cos(40 degrees)

groundspeed^2 = 36100 + 900 - 2 * 190 * 30 * 0.766

groundspeed^2 = 36100 + 900 - 109080

groundspeed^2 = 15320

Taking the square root of both sides:

groundspeed = √(15320)

groundspeed ≈ 123.8 mph

Rounding to the nearest whole number:

groundspeed ≈ 124 mph

Therefore, the groundspeed of the plane, rounded to the nearest whole number, is 124 mph.