Question

The distance that a person can see to the horizon on a clear day from a point above the surface of Earth varies directly as the square root of the height at that point. If a person 144 m above the surface of Earth can see 18 km to the horizon, how far can a person see to the horizon from a point 64 m above the surface?

Answer 1

Using the given direct variation relationship where the distance to the horizon varies directly as the square root of the observer's height above the surface, a person 64 m above the surface can see approximately 12 km to the horizon.

Step-by-step explanation:

The question asks us to determine the distance to the horizon from a point 64 m above the surface of the Earth, given that at a height of 144 m the horizon is 18 km away. This relies on the mathematical relationship where the distance to the horizon varies directly as the square root of the height above the Earth's surface. Using the given relationship, if the distance D varies directly as the square root of the height h, we can represent this as D = k * √h, where k is the constant of proportionality.

Given that the distance to the horizon at 144 m is 18 km, we can first find the constant k by solving for k in D = k *√h. Substituting the given values:

• 18 km = k * √(144 m)

Thus, k can be calculated as follows:

• k = 18 km / √(144 m)
• k = 18 km / 12 m
• k = 1.5 km/m½

Now, using the value of k, we can find the distance to the horizon from 64 m above the Earth's surface. Let D' be this distance:

• D' = 1.5 km/m½ * √(64 m)
• D' = 1.5 km/m½ * 8 m
• D' = 12 km

Therefore, a person standing at a height of 64 m above the Earth's surface can see approximately 12 km to the horizon.