Problem 43 - Precalculus Essentials

Question

The Statue of Liberty is approximately 305 feet tall. If the angle of elevation from a ship to the top of the statue is \(23.7^{\circ}\) how far, to the nearest foot, is the ship from the statue's base?

Step-by-Step Solution

Answer

The ship is approximately 721 feet from the base of the Statue of Liberty.

Step 1: Identify Known Variables

We know that the height of the Statue of Liberty \(H\) is 305 ft and the angle of elevation \(A\) is \(23.7^{\circ}\). The distance of the ship from the base of the Statue, \(D\), is what we need to determine.

Step 2: Setup the Equation using Tangent

Since in a right triangle, tangent of an angle is equal to the ratio of the side opposite the angle to the side adjacent to the angle, we can setup the equation using the formula: \(\tan(A) = \frac{H}{D}\) which can be rearranged to solve for D: \(D = \frac{H}{\tan(A)}\) .

Step 3: Substitute Known Values and Solve

Substitute \(H = 305\) ft and \(A = 23.7^{\circ}\) into the equation: \(D = \frac{305}{\tan(23.7)}\) . Solving, we get \(D \approx 721\) feet to the nearest foot.

Key Concepts

Tangent Ratio

Understanding the tangent ratio is fundamental in right triangle trigonometry and crucial for solving problems like the one involving the Statue of Liberty.

The tangent ratio, in a right triangle, concerns the angle of interest (other than the right angle). It is the ratio of the length of the side opposite the angle to the length of the side adjacent to this angle. Mathematically, for an angle, let's say \( \theta \), the tangent is defined as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

What makes the tangent function so useful is its application in situations where you have a direct line of sight to an object, such as looking up at a tall building or, as in our exercise, the Statue of Liberty. By measuring the angle of elevation, you can use the tangent ratio to find the distance to the object—the side adjacent to the angle—assuming you have the height of the object—the side opposite the angle.

Right Triangle Trigonometry

Right triangle trigonometry deals with the relationships between the angles and the sides of a right triangle. When we reference the tangent ratio in the context of a right triangle, we're using one of the three primary trigonometric functions, alongside sine and cosine.

In our Statue of Liberty problem, we specifically utilized the fact that we're dealing with a right-angled triangle formed by the line of sight from the ship to the top of the statue and the statue's height and base. We harnessed the tangent of the angle of elevation to find the distance from the ship to the statue's base.

This process illustrates a broader theme in right triangle trigonometry; given a single angle and one side, you can determine the remaining sides using trigonometric ratios. Clear understanding and application of these functions enable us to solve real-world problems that otherwise would be difficult to tackle with classical geometric tools alone.

Problem-Solving in Precalculus

Problem-solving in precalculus often entails translating real-world scenarios into mathematical models that can be analyzed and solved. To effectively tackle a problem, like the one involving the Statue of Liberty, follow these key steps:

• Determine what you know and what you need to find out.
• Choose the appropriate mathematical tools or concepts required to address the problem.
• Translate the real-world scenario into a solvable equation using those tools.
• Carry out the manipulation and calculation to arrive at a solution.
• Interpret the solution in the context of the original problem.

By outlining the steps, not only do you create a structured plan of action, but you also increase the likelihood of detecting and eliminating possible errors at early stages. Moreover, it encourages a deep comprehension of the underpinning mathematical principles, rather than simply performing routine or memorized procedures.