porter wants to fly on a broom to luxtown, 1500 [S10degE] from where he lives now. though, there is a wind of 42km/h [W] and his broom has airspeed 25m/s. find angle he should point his broomstick, his velocity and flight time

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Porter wants to fly his broom to Luxtown, which is 1500 kilometers away in the direction South 10 degrees East from his current location. However, there's a westward wind blowing at 42 kilometers per hour, and his broom has an airspeed of 25 meters per second. We need to find the heading angle he should point his broomstick, his actual ground speed, and the total flight time. First, we convert the broom's airspeed from 25 meters per second to 90 kilometers per hour. Next, we set up a coordinate system where South is the positive y-axis and East is the positive x-axis. The problem becomes a vector addition: the ground velocity equals the airspeed vector plus the wind velocity vector. The target ground velocity is in the direction South 10 degrees East, the wind velocity is 42 kilometers per hour westward, and the airspeed is 90 kilometers per hour in an unknown direction that we need to find. Now we break down each vector into its x and y components. The target ground velocity has components v_g sine 10 degrees in the x direction and v_g cosine 10 degrees in the y direction. The wind velocity has components negative 42 in the x direction and zero in the y direction. The airspeed has components 90 sine phi in the x direction and 90 cosine phi in the y direction, where phi is the unknown heading angle from South towards East. This gives us two component equations: v_g sine 10 degrees equals 90 sine phi minus 42, and v_g cosine 10 degrees equals 90 cosine phi. Now we solve the system of equations. From the second equation, we can express v_g as 90 cosine phi divided by cosine 10 degrees. Substituting this into the first equation and simplifying, we get sine of phi minus 10 degrees equals seven fifteenths times cosine 10 degrees, which is approximately 0.4596. Solving for phi minus 10 degrees gives us approximately 27.36 degrees, so phi is approximately 37.36 degrees. Therefore, Porter should point his broomstick in the direction South 37.4 degrees East. Now we calculate the ground speed using our found heading angle. The ground speed equals 90 cosine 37.36 degrees divided by cosine 10 degrees, which gives us approximately 72.6 kilometers per hour. Finally, we calculate the flight time by dividing the distance of 1500 kilometers by the ground speed of 72.6 kilometers per hour, resulting in approximately 20.7 hours. To summarize our final answers: Porter should point his broomstick at heading South 37.4 degrees East, his ground speed will be 72.6 kilometers per hour, and the total flight time will be 20.7 hours.