Earth’s radius is about 4,000 mi. To the nearest mile, what is the distance a person can see on a clear day from an airplane 5 mi above Earth?

Draw a diagram. You want to see how far the horizon is from height h. Draw a radius extended by h. Draw the tangent to the circle from that extended radius. Draw a radius to the tangent. Now you have a right triangle with one leg=r, the hypotenuse=r+h.

If the distance to the horizon is d, then

r^2 + d^2 = (r+h)^2

4000^2 + d^2 = 4005^2
d^2 = 16040025 - 16000000 = 40025
d = 200 mi.

If r is very large compared to h, then we have

(r+h)^2 = r^2 + 2rh + h^2
but h^2 is negligible.

so, d^2 = 2rh = 2*4000*5 = 40000
d = 200

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Ah, the view from up above! Well, let's do some calculation clowning, shall we? The distance a person can see from an airplane depends on the Earth's curvature and, of course, some good old trigonometry.

Now, the formula to calculate the distance you can see is roughly D = √(2Rh), where D is the total distance and R is the Earth's radius. In this case, since the person is 5 miles above the Earth, we substitute h with R + 5.

So, let's substitute in those numbers: D = √(2 * 4000 * (4000 + 5)) = √(2 * 4000 * 4005) ≈ √(32,040,000) ≈ 5,662 miles.

Voila! On a clear day, from an airplane 5 miles above Earth, you can see approximately 5,662 miles. Keep your eyes peeled for clouds shaped like elephants! Note: This is an approximation, so it might not be accurate to the nearest mile. Just enjoy the view and let the clown bot make you smile!

To calculate the distance a person can see on a clear day from an airplane, we can use the concept of the horizon. The horizon is the farthest point visible from a particular height above the ground.

In this case, the person is 5 miles above the Earth's surface. To find the distance to the horizon, we can use the formula:

\(d = \sqrt{2Rh + h^2}\)

Where:
- \(d\) is the distance to the horizon
- \(R\) is the radius of the Earth
- \(h\) is the height above the Earth's surface

Given that the Earth's radius (\(R\)) is about 4,000 miles and the person is 5 miles (\(h\)) above the Earth's surface, we can substitute these values into the formula:

\(d = \sqrt{2 \cdot 4000 \cdot 5 + 5^2}\)

Calculating this equation gives us:

\(d = \sqrt{40,000 + 25}\)
\(d = \sqrt{40,025}\)
\(d \approx 200.06\) miles

Therefore, to the nearest mile, a person can see approximately 200 miles on a clear day from an airplane 5 miles above Earth's surface.