You observe a plane approaching overhead and assume that its speed is 550 miles per hour. The angle of elevation of the plane is 16 degrees at one time and 57 degrees one minute later. Approximate the altitude of the plane.

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31,500 ft

To approximate the altitude of the plane, we can use trigonometry and the concept of tangent.

Let's denote the altitude of the plane as "h" (in miles) and the time difference between the two observations as "t" (in minutes).

From the given information, we know the difference in angles of elevation is 57 degrees - 16 degrees = 41 degrees.

We can set up a proportion using the tangent function:

tangent(41 degrees) = h / (550 * (t/60))

Now let's solve for "h":

h = tangent(41 degrees) * 550 * (t/60)

Plug in the given values, t = 1 minute and convert it to hours by dividing by 60:

h = tangent(41 degrees) * 550 * (1/60)

Using a calculator or math software, we can find that tangent(41 degrees) ≈ 0.869.

Substituting this value into the equation:

h ≈ 0.869 * 550 * (1/60)

Simplifying further:

h ≈ 8.114 miles

Therefore, the approximate altitude of the plane is 8.114 miles.