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Chapter 12: Problem 12

Application Dakota Davis uncovers the remains of a square-based Egyptian pyramid. The base is intact and measures 130 meters on each side. The top of the pyramid has eroded away, but what remains of each face of the pyramid forms a \(65^{\circ}\) angle with the ground. What was the original height of the pyramid?

Short Answer

Expert verified
The original height of the pyramid was approximately 139.39 meters.

Step by step solution

01

Understand the Problem

You need to find the original height of a square-based pyramid. The base is intact and measures 130 meters on each side, and the remaining faces form a 65-degree angle with the ground.
02

Visualize the Triangle

Visualize the right triangle formed by the pyramid's height (h), half the base's side length (65 m), and the slanted height of the face.
03

Identify the Right Triangle

The height of the pyramid (h) is one leg, half the base (65 m) is another leg, and the slanted height is the hypotenuse.
04

Use Trigonometric Functions

The angle between the base and the slant height is 65 degrees. Use the tangent function: \(\tan(65^{\circ}) = \frac{h}{65}\)
05

Solve for Height

Rearrange the tangent formula to solve for h: \(h = 65 \tan(65^{\circ})\). Calculate the value.
06

Calculate the Height

Substitute the value of tan(65掳): \(\tan(65^{\circ}) \approx 2.1445\). Now multiply: \(h = 65 \times 2.1445 \approx 139.39\) meters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

right triangle
A right triangle is a type of triangle that has one angle equal to 90 degrees. This means that one of its sides is perpendicular to the other. In this exercise, the height of the pyramid, the half of the base, and the slanted height of the pyramid form a right triangle:
  • The height of the pyramid (h) stands vertical.
  • Half of the base (65 m) is horizontal.
  • The slanted height lies along the face of the pyramid.
Understanding this right triangle is key to solving the problem. Recognize that the height you need to find is one of the legs of this right triangle.
trigonometric functions
Trigonometric functions are essential tools for solving problems involving right triangles. The primary trigonometric functions are sine, cosine, and tangent. In this problem, we focus on the tangent function.
The tangent (tan) of an angle in a right triangle is the ratio of the opposite side to the adjacent side. For this exercise with a 65-degree angle:
  • Opposite side is the height (h) of the pyramid.
  • Adjacent side is half the base, 65 meters.
This relationship can be written as: \(\tan(65^\text{掳}) = \frac{h}{65}\). We use this formula to find the height of the pyramid.
pyramid height calculation
To find the original height of the pyramid, follow these steps:
First, note that we have the angle and one side (half the base). We need the height (h), so we'll use the tangent formula: \( \tan(65^\text{掳}) = \frac{h}{65} \). Rearrange this to solve for h: \( h = 65 \tan(65^\text{掳}) \). Using a calculator, we find that \( \tan(65^\text{掳}) \) is approximately 2.1445. Thus, \( h = 65 \times 2.1445 \), which gives approximately 139.39 meters.
angle of elevation
The angle of elevation in this context is the angle between the base of the pyramid and the slanted height of its face. It鈥檚 given as 65 degrees. This angle is crucial as it connects the height of the pyramid and half of its base through the tangent function. When you look up from any point of the base towards the remnant of the pyramid鈥檚 face, this angle of 65 degrees is what you would measure. It signifies how steep the pyramid's faces were.

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