Question

Egypt's Great Pyramid is \(150 \mathrm{~m}\) tall and has a square base with a length of \(225 \mathrm{~m}\). What is the approximate volume of the pyramid in cubic meters? The formula for the volume \(V\) of a pyramid of base area \(B\) and height \(h\) is given by \(V=\frac{1}{3} B h\). (A) 500,000 (B) \(1,500,000\) (C) \(2,500,000\) (D) \(7,500,000\)

Short answer

(C) $2,500,000$

Step-by-step solution

Identify the given information

Egypt's Great Pyramid has a height of \(150 \mathrm{~m}\) and a square base with a side length of \(225 \mathrm{~m}\).

Find the base area of the pyramid

Since the base of the pyramid is a square, we can calculate the base area \(B\) using the formula \(B = s^2\), where \(s\) represents the side length. With a side length of \(225 \mathrm{~m}\), we get: \[B = (225 \mathrm{~m})^2 = 50625 \mathrm{m}^2\]

Calculate the volume of the pyramid using the given formula

Now, let's use the formula for the volume of a pyramid \(V = \frac{1}{3}Bh\). We already know the base area \(B=50625 \mathrm{m}^2\) and the height \(h=150 \mathrm{~m}\), so we can substitute these values into the formula: \[V = \frac{1}{3}(50625 \mathrm{m}^2)(150 \mathrm{~m})\]

Solve for the volume

Now we need to multiply and simplify the expression: \[V = \frac{1}{3}(7593750 \mathrm{m}^3)\] \[V \approx 2531250 \mathrm{m}^3\]

Choose the closest answer from the given options

The volume we calculated is approximately \(2,531,250~\mathrm{m}^3\). Looking at the given options, the closest answer is choice (C) \(2,500,000\).

Key concepts

Egypt's Great Pyramid

The Great Pyramid of Giza is one of the most famous monuments in the world and a marvel of ancient engineering. It was constructed over 4,500 years ago during the reign of Pharaoh Khufu. The pyramid originally stood at about 146.6 meters tall, but with erosion and the loss of its outer casing, it now measures approximately 138.5 meters. For mathematical purposes, we often use more standardized dimensions.

• Height: Typically rounded to about 150 meters in exercises.
• Base: Square with a side length of about 225 meters.

Having these standardized dimensions helps in performing calculations concerning its volume or other geometrical properties.

Pyramid Volume Formula

The volume of a pyramid can be calculated using its base area and height. The formula is simple yet powerful:

• Formula: \( V = \frac{1}{3} B h \)
• Where:
• \(B\) = Base area of the pyramid
• \(h\) = Height of the pyramid

This formula applies to any pyramid, not just square-based ones. It shows why ancient builders had to have a strong grasp of mathematics to construct such complex structures like Egypt’s pyramids. Calculating volume is essential for understanding the amount of materials needed and for structural considerations.

Geometry Skills

Understanding geometry is crucial when tackling problems about pyramids. Firstly, you need to know how to find areas of different shapes. For Egypt's Great Pyramid, since the base is a square:

• Base Area (\(B\)): Calculated using \(B = s^2\), where \(s\) is the side length. In our problem, \(s = 225 \,\mathrm{m}\).
• Once you find the base area, you use it in volume calculations.

These skills help build a deeper understanding of shapes and their properties. The exercises in geometry improve your spatial awareness and teaches you how to tackle real-world problems logically.

Mathematical Calculations

To solve the problem of finding the volume of Egypt's Great Pyramid, we follow a series of calculations:

• Step 1: Calculate the area of the square base: \( B = (225 \,\mathrm{m})^2 = 50625 \,\mathrm{m}^2 \).
• Step 2: Apply the volume formula: \( V = \frac{1}{3} (50625 \,\mathrm{m}^2) (150 \,\mathrm{m}) \).
• Step 3: Calculate the result: First multiply 50625 by 150, then divide by 3, resulting in \( V \approx 2531250 \,\mathrm{m}^3 \).
• Conclusion: Compare with given choices: The closest option is (C) 2,500,000.

Each step involves basic but essential mathematical operations: multiplication, division, and using formulas. These skills underscore the importance of precision and accuracy in calculations. Having strong calculation skills will simplify solving more complex problems in the future.