In Exercises 45-56, set up appropriate systems of two linear...

In Exercises $45-56,$ set up appropriate systems of two linear equations in two unknowns and then solve the systems by determinants. All numbers are accurate to at least two significant digits. A surveyor measures the angle of elevation to the top of a hill to be $15.5^{\circ} .$ He then moves $345 \mathrm{ft}$ closer, on level ground, and remeasures the angle of elevation to be $21.4^{\circ}$ (see Fig. 5.36 ). Find the height $h$ of the hill and the distance $d .$ Hint: Use the tangent function to set up a system of equations.

In Exercises $45-56,$ set up appropriate systems of two linear equations in two unknowns and then solve the systems by determinants. All numbers are accurate to at least two significant digits.
A surveyor measures the angle of elevation to the top of a hill to be $15.5^{\circ} .$ He then moves $345 \mathrm{ft}$ closer, on level ground, and remeasures the angle of elevation to be $21.4^{\circ}$ (see Fig. 5.36 ). Find the height $h$ of the hill and the distance $d .$ Hint: Use the tangent function to set up a system of equations.

Key Concepts

Systems of Linear Equations

This concept involves setting up and solving two or more equations that include the same unknowns, where each equation represents a different relationship between the variables. In applications such as surveying or physics, systems of linear equations allow for the determination of unknown measurements by leveraging the relations derived from various observations or constraints.

Determinants

Determinants are scalar values computed from square matrices that are critical in solving systems of linear equations, especially via methods like Cramer's Rule. They help determine whether a unique solution exists, and when the determinant is nonzero, they facilitate an efficient method for finding the values of unknown variables.

Tangent Function in Trigonometry

The tangent function, defined as the ratio of the opposite side to the adjacent side in a right triangle, is pivotal for relating angles to distances. It is widely used in problems involving angles of elevation or depression, enabling the formulation of relationships between unknown distances and heights in a variety of practical contexts.

Trigonometric Modeling in Real-World Applications

Trigonometric modeling uses functions such as the tangent to translate physical scenarios into mathematical equations. This approach is particularly useful in fields like surveying, where measurements of angles need to be converted into concrete dimensions like distances and heights, providing a systematic way to analyze and solve spatial problems.

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