Problem 8 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

The pyramid of Cheops, the largest of the Egyptian pyramids, is 146.6 meters tall with a square base of side 230.4 meters. What is its volume?

Step-by-Step Solution

Answer

The volume of the pyramid is approximately 2592303.47 m³.

Step 1: Calculate Base Area

Start by calculating the area of the square base of the pyramid. The area of a square is given by the formula: \[ A = s^2 \] where \( s \) is the side length of the square. Here, \( s = 230.4 \) meters, so:\[ A = (230.4)^2 \] Calculation: \[ A = 230.4 \times 230.4 = 53041.76 \text{ m}^2 \]

Step 2: Apply Pyramid Volume Formula

Next, use the formula for the volume of a pyramid: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] We've already calculated the base area as 53041.76 m², and the given height of the pyramid is 146.6 meters. Use these values in the formula:\[ V = \frac{1}{3} \times 53041.76 \times 146.6 \]

Step 3: Compute Volume

Perform the multiplication to find the volume:\[ V = \frac{53041.76 \times 146.6}{3} \]First, multiply the base area by the height:\[ 53041.76 \times 146.6 = 7776910.416 \]Now, divide by 3 to get the volume:\[ V = \frac{7776910.416}{3} = 2592303.472 \text{ m}^3 \]

Step 4: Final Volume

Therefore, the volume of the pyramid is approximately 2592303.47 cubic meters.

Key Concepts

Pyramid Volume

Calculating the volume of a pyramid is an interesting and straightforward process, especially when you have all the key dimensions. A pyramid is a three-dimensional geometric shape with a base that can be any polygon and triangular faces that converge to a single point, known as the apex. The volume of a pyramid is quite unique. It depends on both the area of its base and its height. The formula for calculating the volume is given by:

• \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \)

For example, to find the volume of the pyramid of Cheops, we take its base area and multiply it by the height, then divide the result by three. This final step of dividing by three accounts for the pyramid's tapered structure compared to a prism with the same base and height, making it simpler and less voluminous. Applying this formula helps you understand how different geometrical structures compare in terms of space they occupy.

Area of Square

Understanding how to calculate the area of a square is a fundamental skill in geometry. A square is a special quadrilateral where all four sides are equal in length, and all angles are right angles. Calculating the area of a square is simple and can be done using the formula:

• \( A = s^2 \)

where \( s \) is the length of one side of the square. This formula works because a square's area is just the length of one side multiplied by itself. In the example of the pyramid of Cheops, the base is a square with a side length of 230.4 meters. To find its area, multiply 230.4 by itself, which results in 53041.76 square meters. This area then plays a crucial role in further calculations, particularly in determining the pyramid's volume.

Formula Application

Applying formulas in mathematical problems requires understanding each component and its role in the solution. In the context of calculating the pyramid's volume, knowing how to apply the formula is vital. The key steps involve:

• Identifying the necessary dimensions (e.g., the base area and height).
• Plugging these values into the correct formula.
• Performing the calculations methodically to reach the correct answer.

In our pyramid example, once the base area was calculated to be 53041.76 square meters and knowing the height is 146.6 meters, we inserted these into the volume formula. Multiplying the base area by the height yields a preliminary product, which is then divided by three to account for the pyramid's shape. Understanding this procedure ensures accurate results and enhances competency in solving similar problems.

Mathematical Calculations

Performing accurate mathematical calculations is key to solving geometric problems effectively. This involves correctly executing arithmetic operations and ensuring precision throughout the process. In calculating the pyramid's volume, the steps were:

• Multiplying the base area (53041.76 m²) by the height (146.6 m).
• Calculating the product as 7776910.416.
• Dividing this result by three to obtain the final volume of 2592303.472 cubic meters.

These calculations must be done carefully using a calculator or software tool to ensure precision, especially with large and complex numbers. Keeping track of each step and verifying results help in reducing errors and understanding the relationship between geometric measures and numerical accuracy.