Problem 42 - Algebra 2

Question

WASHINGTON MONUMENT The Washington Monument, in Washington, D.C. is 555 feet 5\(\frac{1}{8}\) inches tall and weighs \(90,854\) tons. The monument is topped by a square aluminum pyramid. The sides of the pyramid's base measure 5.6 inches, and the pyramid is 8.9 inches tall. Estimate the slope that a face of the pyramid makes with its base.

Step-by-Step Solution

Answer

The slope of the pyramid's face is approximately 72.5°.

Step 1: Understand the Slope of the Pyramid

The slope of a pyramid's face with its base can be determined using trigonometry. Specifically, we are interested in the angle between the inclined face of the pyramid and the horizontal base.

Step 2: Extract Known Values

The base of the square pyramid has sides measuring 5.6 inches, and its height is 8.9 inches. These values will help us compute the slant height of the pyramid, which is needed to find the slope.

Step 3: Compute the Slant Height

The slant height (\( l \) ) of the pyramid can be found using the Pythagorean theorem. First, calculate the half diagonal of the base. The diagonal (\( d \) ) of the square base is \( d = 5.6\sqrt{2} \), so half the diagonal is \( \frac{5.6\sqrt{2}}{2} \). Using the Pythagorean theorem, \( l = \sqrt{\left(\frac{5.6\sqrt{2}}{2}\right)^2 + 8.9^2} \).

Step 4: Solve for the Slant Height

Calculate the value of the slant height: \( l = \sqrt{\left(\frac{5.6\sqrt{2}}{2}\right)^2 + 8.9^2} \) \( \approx \sqrt{28.56} \approx 5.34 \) inches.

Step 5: Determine the Slope of the Pyramid

The slope of the pyramid's face can be found using the tangent of the angle \( \theta \):\( \tan \theta = \frac{\text{height of the pyramid}}{\frac{\text{base of the pyramid}}{2}} \).In this case, \( \tan \theta = \frac{8.9}{\frac{5.6}{2}} \).

Step 6: Calculate the Angle

Solve for the angle \( \theta \) using the inverse tangent:\( \theta = \arctan \left( \frac{8.9}{2.8} \right) \approx \arctan(3.18) \approx 72.5^{\circ} \).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that focuses on the relationships between the sides and angles of triangles. It's quite useful in various fields such as engineering, physics, and even in everyday problem-solving. In the context of the Washington Monument exercise, trigonometry helps us determine the angle of inclination for the pyramid at the top of the monument.
The exercise uses a set of known values related to the pyramid's dimensions to find this angle. By understanding trigonometric ratios like sine, cosine, and tangent, we can efficiently solve for unknowns in a right triangle. These ratios relate the angle to the ratios of the sides of the triangle. Therefore, once we have the measurements related to the structure, we can calculate angles like the slope of the pyramid's face.
So, through trigonometry, we transform a geometric problem into numerical values that are much easier to compute and understand.

Slant Height

The slant height of a pyramid is the distance from the top of the pyramid (the apex) down the middle of the face to the base. In simple terms, it's the diagonal height along the slanted surface rather than directly down to the center of the base.
For the pyramid on the Washington Monument, calculating the slant height allows us to understand the exact shape and incline of the pyramid's face. To do this, we employ the Pythagorean theorem.
- We begin by calculating the diagonal of the square base given its side length.
- Then, we take half of this diagonal to build our right triangle with the pyramid's vertical height.
- Applying the Pythagorean theorem, we solve for the hypotenuse, which is the slant height.
Understanding the slant height is crucial because it connects directly to many engineering principles and is essential for accurate architectural schemes.

Tangent of an Angle

The tangent of an angle in a right triangle is a fundamental concept in trigonometry. It's defined as the ratio of the opposite side to the adjacent side of the angle in question. This ratio helps us understand how steep or shallow a particular angle is.
In this exercise, you find the tangent of the angle at which the pyramid faces incline relative to its base. We identify the opposite side as the vertical height of the pyramid and the adjacent side as half the base's length. Knowing these, we set up the equation for tangent:
\[\tan \theta = \frac{\text{height of the pyramid}}{\frac{\text{base of the pyramid}}{2}}\]
Finding this ratio is the step before using other trigonometric methods such as solving for angles.

Inverse Tangent

Inverse tangent, often written as \(\arctan\), is a function used to find an angle when you know its tangent. In practical terms, it's the reverse of the tangent function.
In this particular problem, once you have calculated the tangent of the pyramid's angle using its height and base, inverse tangent is used to uncover the actual degree measurement of the angle between the face of the pyramid and its base.
After finding the tangent ratio as about 3.18, the inverse tangent will provide the angle:
\[\theta = \arctan(3.18) \approx 72.5^\circ\]
This angle tells us the steepness of the pyramid's surface, crucial for understanding the structure's design.
Mastery of inverse trigonometric functions like the inverse tangent broadens your ability to solve problems involving angles and triangles.