Question

The Great Pyramid in Egypt has a height of approximately 150 meters and a base of 50,000 square meters. In cubic meters, what is its approximate internal volume? A. \(7,500,000\) B. \(5,000,000\) C. \(2,500,000\) D. 750,000

Short answer

The approximate internal volume of the Great Pyramid is \(2,500,000\) cubic meters. Therefore, the correct answer is (C) \(2,500,000\).

Step-by-step solution

Recall the formula for the volume of a pyramid

The formula for the volume of a pyramid is V = (1/3) * base_area * height, where V represents the volume, base_area represents the area of the base, and height represents the height of the pyramid.

Write down the values given in the problem

We are given the height of the Great Pyramid: 150 meters and the base area: 50,000 square meters.

Substitute the given values into the formula and solve for V

We can now substitute the given values into the formula: V = (1/3) * 50,000 * 150. To find the volume, we can first multiply 50,000 by 150, which gives us 7,500,000. Then, we can divide 7,500,000 by 3 (since the formula has a fraction of 1/3), which gives us: V = 2,500,000.

Interpret the result

The approximate internal volume of the Great Pyramid is 2,500,000 cubic meters. Therefore, the correct answer is (C) \(2,500,000\).

Key concepts

Mathematical Formulas

To find the volume of a pyramid, it is essential to use the correct mathematical formula. The volume formula for a pyramid is: \[V = \frac{1}{3} \times \text{base_area} \times \text{height}\]This formula shows that the volume of a pyramid depends on two crucial measurements: the area of its base and its height. It’s important to understand how fractions play a role in the formula. Multiplying the area of the base by the height gives the volume of a prism with the same base area and height. Since a pyramid is essentially a prism that tapers to a point, its volume is a third of the equivalent prism's volume.
By understanding and applying this formula, you can solve for the volume of any pyramid shape, as long as you know the base area and height.

Geometry

Geometry provides the shape and dimension of objects which is fundamental in understanding solids like pyramids. The Great Pyramid of Egypt is an iconic example of a square-based pyramid, which means its base is a square. To find the base area of a square, you simply multiply its side length by itself. In practice, measurements like height, which is the perpendicular distance from the base to the apex, are crucial in determining the pyramid's volume.

• Base Area: If given directly, like the 50,000 square meters in this case, use it in your calculations.
• Height: This is the vertical distance from the base to the point directly above the center of the base.

It is essential to conceptualize how the base and height align with one another in three-dimensional space to ensure accurate calculations. Understanding the geometric properties helps visualize and compute these values correctly when applying the volume formula.

Volume Calculation

Calculating the volume of a pyramid involves substituting known values into the volume formula and performing simple arithmetic operations. Given the base area and height, as in the Great Pyramid example, you substitute these into the formula \[V = \frac{1}{3} \times 50,000 \times 150\]The calculation involves first multiplying the base area (50,000 square meters) by the height (150 meters) which gives 7,500,000 cubic meters. Then, applying the final step of the formula, dividing by 3, simplifies it to the actual pyramid volume: 2,500,000 cubic meters.

Steps to ensure accuracy:

• Ensure your base area and height are in compatible units, typically both in meters for such problems.
• Multiply these values to get a preliminary result.
• Lastly, divide by 3 reflecting the tapering shape of the pyramid to find its volume.

By following these steps, calculating the volume of a pyramid becomes a straightforward operation. Understanding these calculations enables you to solve similar geometry problems with confidence.