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The distance to horizon formula is a vital concept when calculating how far a person can see from a certain height. It is derived from geometric principles related to the curvature of the Earth. The formula used is:

• \( d = \sqrt{\frac{3h}{2}} \)

Here, \(d\) is the distance to the horizon in miles, and \(h\) is the height in feet from which you are viewing. This formula assumes that the Earth is perfectly spherical and does not account for any atmospheric refraction. It simplifies complex trigonometric calculations into an accessible form.
By substituting the height into the formula, you can easily determine how far you might be able to see over the horizon. It is important to understand this formula as it lays the groundwork for practical applications like navigation and observing natural surroundings.

When solving mathematical problems, expressing answers in simplified radical form can provide a more accurate representation than approximate decimal values. The goal is to simplify the radical expression as much as possible.
For the exercise given, once you've substituted the height, you simplify \(\sqrt{180}\) to \(6 \sqrt{5}\). Here's how:

• Factor 180 into its prime factors, giving you \(36 \times 5\).
• Recognize that 36 is a perfect square and can be factored out of the radical, resulting in \(6\sqrt{5}\).

This simplified radical form is exact and useful in maintaining accuracy before moving to decimal approximations. Simplification often reveals underlying patterns and relationships in mathematical expressions.

The substitution method is crucial in algebra for replacing variables with concrete numbers to solve problems or evaluate expressions. In this context, it involves substituting the height of the ship's star deck into the distance formula.
Given that the height \(h\) is 120 feet, you replace \(h\) in the original formula \(d=\sqrt{\frac{3h}{2}}\) with 120. This transforms the equation to \(d=\sqrt{\frac{3 \times 120}{2}}\). A clear substitution simplifies the computing process and helps in deriving solutions systematically.
Substitution is a foundational algebraic skill that is widely applicable in various branches of mathematics and real-world problem-solving scenarios.

Once you have an expression in a simplified radical form, converting it into a decimal approximation is often necessary for practical use. This final step can involve using a calculator for precision.
With \(6\sqrt{5}\) as your simplified radical form, squaring and simplifying it gives you an approximate decimal value. Here, it calculates to \(d \approx 13.4\) miles.
Decimal approximations are crucial for making numerical results easily understandable, especially in real-world applications where exact radical forms are impractical. They bridge the gap between the mathematical precision of radicals and the usability of numerical values in everyday scenarios.