Question

the circle at the right represents a planet. the radius of the planet is about 5300 km. find the distance d to the horizon that a person can see on a clear day from the following height h above the planet.

h = 4 km

a person can see a distance of approximately km on a clear day from a height of 4 km above the given planet.
(round to the nearest tenth as needed.)

Explanation:

Step1: Apply the Pythagorean theorem

The radius \( r = 5300\) km, height \( h=4\) km. The distance from the center of the planet to the person is \(r + h=5300 + 4=5304\) km. Let the distance to the horizon be \(d\). By the Pythagorean theorem \(d=\sqrt{(r + h)^{2}-r^{2}}\).

Step2: Expand the formula

$$
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Substitute \(r = 5300\) km and \(h = 4\) km into the formula: \(d=\sqrt{2\times5300\times4+4^{2}}=\sqrt{42400 + 16}=\sqrt{42416}\).

Step3: Calculate the value

\(\sqrt{42416}\approx205.95\approx206.0\) (rounded to the nearest tenth).

Answer:

\(206.0\)