Problem 43 - Thinking Mathematically

Question

The Great Pyramid outside Cairo, Egypt, has a square base measuring 756 feet on a side and a height of 480 feet. a. What is the volume of the Great Pyramid, in cubic yards? b. The stones used to build the Great Pyramid were limestone blocks with an average volume of \(1.5\) cubic yards. How many of these blocks were needed to construct the Great Pyramid?

Step-by-Step Solution

Answer

a. The volume of the Great Pyramid is \(3,370,880\) cubic yards. b. Approximately \(2,247,253\) limestone blocks were needed to construct the Great Pyramid.

Step 1: Convert measurements to yards

First, convert all measurements from feet to yards because the volume of the limestone blocks is given in cubic yards. There are \(3\) feet in a yard, so: \(756 \text{ feet} = \frac{756}{3} = 252 \text{ yards}\) \(480 \text{ feet} = \frac{480}{3} = 160 \text{ yards}\)

Step 2: Calculate the volume of the pyramid

Next, consider the formula for the volume of a pyramid, \(V = \frac{1}{3}bh\), where \(b\) is the area of the base and \(h\) is the height. Given that the pyramid has a square base, \(b = side^{2} = 252^{2} = 63504 \text{ square yards}\). Substituting these values into the formula, \(V = \frac{1}{3} * 63504 * 160 = 3,370,880 \text{ cubic yards}\)

Step 3: Calculate the number of limestone blocks

Finally, divide the total volume of the pyramid by the volume of a limestone block to determine the number of blocks. This is: \(blocks = \frac{3,370,880}{1.5} = 2,247,253.33\). However, there can't be fractional blocks, so round to the nearest whole number which is: 2,247,253 blocks.

Key Concepts

Geometry in Mathematics

Geometry is a fundamental branch of mathematics that deals with the properties and relationships of shapes, sizes, and configurations of figures. The study of three-dimensional shapes such as pyramids, which are polyhedrons with a polygon base and triangular faces meeting at a common point called the apex, is an essential part of geometry.

Understanding the properties of these geometric shapes not only allows students to appreciate architectural feats like the Great Pyramid of Giza but also equips them with the analytical skills to solve problems in various fields such as engineering, architecture, and even physics.

Application of Geometric Understanding

When calculating the volume of a pyramid, a clear apprehension of its shape, the relative positions of its faces, and the angle between the base and the sides are crucial. The use of geometric formulas to determine volume is a direct application of spatial reasoning – a skill honed through the study of geometry.

Converting Measurements

Mathematics often requires converting measurements from one unit to another. This skill is vital in ensuring accuracy in calculations and making computational results relatable to real-world dimensions.

In the case of the Great Pyramid’s volume calculation, converting the linear measurements from feet to yards is necessary because the volume of limestone blocks is provided in cubic yards. To proficiently execute conversions, one must understand the relationships between units: for instance, knowing that one yard equals three feet.

Mastering Unit Conversion

Mastering unit conversion is essential because it enables students to seamlessly work with different measurement systems. This capability is not only crucial in mathematical problems but also in daily life, science experiments, and technical projects where measurements must be accurately interpreted and utilized.

Pyramid Volume Calculation

The calculation of a pyramid's volume is a specific example of applied geometry. The volume refers to the amount of three-dimensional space the pyramid occupies, and is a measure vital in various practical endeavors, from construction to archeology.

To calculate the volume of a pyramid with a square base, like the Great Pyramid of Giza, the standard formula is: \( V = \frac{1}{3}bh \), where \( b \) is the area of the base and \( h \) is the height. The area of a square base is found by squaring the length of one of its sides (\( side^2 \) ). After finding the area, this value is then multiplied by the height of the pyramid, and the product is divided by three to find the pyramid's volume.

Real-World Significance of Volume Calculation

Volume calculations are not just academic exercises; they have real-world significance. For example, knowing the volume of the Great Pyramid enables historians and architects to estimate the number of blocks used in its construction. For tasks like resource allocation, space optimization, and even in determining storage capacities, the ability to calculate the volume of a pyramid is both practical and invaluable.