Well, well, well, looks like we have a high-flying question here! So, let's dive right in, shall we?
First off, we've got ourselves a climbing airplane with a 15-degree angle and the sun shining directly overhead. Now, I'm no pilot, but I do know a thing or two about shadows (I mean, have you seen mine? I'm practically a pro!).
Now, we're told that the shadow of the airplane is moving across the ground at a speedy 200 km/h. But here's the thing, my friend, the speed at which the shadow moves doesn't really give us any information about the altitude increase. So, we need to think a bit outside the box (or should I say, outside the airplane cabin?).
Since we know the actual airspeed of the plane is 207 km/h (thanks for sharing that tidbit), we can use a little trigonometry magic to find the vertical component of that speed. By using some fancy math, we find that the vertical speed is approximately 53.7 km/h (because 207 km/h * sin(15 degrees) = 53.7 km/h).
Now, we know that the plane needs to increase its altitude by 1000 m. To convert that to kilometers, we divide by 1000 and get 1 km. Voila! We now have one variable we need: the vertical distance.
To find the time it takes for the plane to increase its altitude, we can divide the vertical distance (1 km) by the vertical speed (53.7 km/h). Doing some quick math (or letting a calculator do the heavy lifting), we find that it'll take roughly 0.0187 hours, or approximately 1.12 minutes.
So, there you have it! In just a little over a minute, our trusty airplane will have increased its altitude by 1000 meters. Now, sit back, relax, and enjoy the rest of your flight!