Solve each problem. Distance to the HorizonThe 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?

Verified step by step guidance

1. Identify the relationship: The distance to the horizon varies directly as the square root of the height, which can be expressed as \( d = k \sqrt{h} \), where \( d \) is the distance to the horizon, \( h \) is the height, and \( k \) is the constant of proportionality.
2. Use the given information to find the constant \( k \): Substitute \( d = 18 \) km and \( h = 144 \) m into the equation to get \( 18 = k \sqrt{144} \).
3. Solve for \( k \): Calculate \( \sqrt{144} \) and then divide \( 18 \) by this value to find \( k \).
4. Set up the equation for the new height: Use the equation \( d = k \sqrt{h} \) with the newly found \( k \) and \( h = 64 \) m.
5. Solve for the new distance \( d \): Calculate \( \sqrt{64} \) and multiply by \( k \) to find the distance to the horizon from 64 m above the surface.

Key Concepts

Direct Variation

Direct variation describes a relationship between two variables where one variable is a constant multiple of the other. In this context, the distance to the horizon varies directly with the square root of the height above the Earth's surface. This means that if the height increases, the distance to the horizon increases proportionally, allowing us to set up a ratio to solve for unknown distances.

Square Root Function

The square root function is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. In this problem, the distance to the horizon is proportional to the square root of the height, indicating that as height increases, the increase in distance is not linear but rather follows a square root curve, which grows more slowly as height increases.

Proportional Relationships

Proportional relationships are mathematical expressions that show how two quantities change in relation to each other. In this scenario, we can express the relationship between height and distance to the horizon using a proportionality constant derived from the given data. This allows us to calculate the distance for different heights by maintaining the same ratio established by the initial conditions.