Question

MODELING WITH MATHEMATICS You are standing on the Grand View Terrace viewing platform at Mount Rushmore, 1000 feet from the base of the monument. Not drawn to scale 24° b 1000 ft a. You look up at the top of Mount Rushmore at an angle of 24°. How high is the top of the monument from where you are standing? Assume your eye level is 5.5 feet above the platform. b. The elevation of the Grand View Terrace is 5280 feet. Use your answer in part (a) to fi nd the elevation of the top of Mount Rushmore.

Short answer

The height of Mount Rushmore from where the observer is standing can be calculated by using the tan function and knowing the distance from the observer to the base of the monument. The elevation of the top of Mount Rushmore can be calculated by adding this height to the elevation of the Grand View Terrace.

Step-by-step solution

Use Tangent Function To Find The Height Of The Monument

The tangent of an angle in a right triangle is the length of the opposite side divided by the length of the adjacent side. We can use this property to find the height of the monument (b), using the equation: \[ \tan(24°) = \frac{b}{1000} \]. This can be re-arranged to: \[ b = 1000 * \tan(24°) \].

Calculate The Height Of The Monument

Substituting 24° into the equation, we calculate \( b = 1000 * \tan(24°) \) to find the height of the monument.

Total Height From The Platform

To find the total height from the platform to the top of Mount Rushmore, we need to add the observer's eye level to the calculated height of the monument. Given the observer’s eye level is 5.5 feet, the actual height of the monument from where the observer is standing becomes \( b_{total} = b + 5.5 \). Calculate \( b_{total} \) using the value of \( b \) obtained from Step 2.

Find The Elevation Of The Top Of Mount Rushmore

To find the elevation of the top of Mount Rushmore, we add the total height from the platform to the elevation of Grand View Terrace (5280 feet). The resultant sum is the elevation of the top of Mount Rushmore i.e. \( Elevation_{Rushmore} = Elevation_{Terrace} + b_{total} \). Calculate \( Elevation_{Rushmore} \) using the value of \( b_{total} \) obtained from Step 3.

Key concepts

Tangent function

In trigonometry, the tangent function (\( \tan \)) is a valuable tool, especially when working with right triangles. It describes the ratio between the opposite side and the adjacent side of a right triangle concerning a given angle. For instance, if you have an angle of 24° in a right triangle, the tangent of this angle helps you find unknown side lengths if one side is known. The equation becomes:

• \( \tan(24°) = \frac{\text{opposite side}}{\text{adjacent side}} \)

In real-world situations, the tangent function is often used to calculate heights and distances that are not easily measurable directly. In our example, to find how high the top of Mount Rushmore is from the standing point at the Grand View Terrace, the tangent of the angle viewed is multiplied by the known distance, allowing us to solve for the height using:

• \( b = 1000 \times \tan(24°) \)

This illustrates the practical application of trigonometric functions in actual physical scenarios.

Right triangle properties

Right triangles are fundamental in geometry due to their distinct properties. They consist of one right angle (90°), with the other two angles adding up to 90°. These properties allow mathematicians to establish relationships between the sides using trigonometric ratios like sine, cosine, and tangent.

When dealing with a problem such as finding the height from Mount Rushmore's base to its top, a right triangle can be formed by considering:

• the base as one side,
• the height of the towering structure as the opposite side, and
• the line of sight from your eye to the monument's top as the hypotenuse.

By using these properties, you can apply trigonometric functions to find unknown measures. In our problem, we isolated the height using tangent, which suited perfectly since we had one angle and the distance (adjacent side) known.

Mathematical modeling

Mathematical modeling involves creating a mathematical representation of a real-world situation to solve problems or predict outcomes. In the context of trigonometry, it's about using known mathematical theories to draw meaningful conclusions in practical circumstances.

In the Mount Rushmore example, mathematical modeling comes into play when defining the scenario with a right triangle based on the angle of elevation and the distance from the monument. You utilize formulas like \( \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \) in a structured approach to find values that aren't immediately apparent.

This model, initiated from your perspective at Grand View Terrace, helps compute the height of Mount Rushmore. By adding together the calculated height and known elevations, you deduce its exact elevation top. Thus, mathematical modeling translates abstract concepts into tangible results.

Trigonometric equations

Trigonometric equations allow us to solve problems involving angles and side lengths in triangles. These equations are crucial in defining relationships and solving real-life geometry problems.

In scenarios like the one involving Mount Rushmore, establishing trigonometric equations is key. Using the equation \( \tan(24°) = \frac{b}{1000} \), the height of the statue (\( b \)) can be determined, provided that the tangent value and side length are known.

Solutions involve manipulating these equations to isolate variables of interest. Critical steps include substituting values into the equation, rearranging to solve for the unknown:

• \( b = 1000 \times \tan(24°) \)
• Add observer's eye level: \( b_{total} = b + 5.5 \)
• Final elevation: \( Elevation_{Rushmore} = Elevation_{Terrace} + b_{total} \)

Ultimately, these equations convert theoretical components into practical measurements, enabling a comprehensive understanding of the site's geographical dynamics.