Problem 32 - Basic Technical Mathematics with Calculus

Question

\(\text {Solve the given problems.}\) The volume of a frustum of a pyramid is \(V=\frac{1}{3} h\left(a^{2}+a b+b^{2}\right)\) (see Fig. 2.123 ). (This equation was discovered by the ancient Egyptians.) If the base of a statue is the frustum of a pyramid, find its volume if \(\quad a=2.50 \mathrm{m}, b=3.25 \mathrm{m}, \quad\) and \(h=0.750 \mathrm{m}\)

Step-by-Step Solution

Answer

The volume of the frustum is approximately 6.234 m³.

Step 1: Understand the Formula

The formula to find the volume of a frustum of a pyramid is given as \( V = \frac{1}{3} h(a^2 + ab + b^2) \). Here, \( h \) is the height of the frustum, and \( a \) and \( b \) are the lengths of the sides of the two parallel square bases.

Step 2: Substitute the Given Values

We are given the values: \( a = 2.50 \, \mathrm{m} \), \( b = 3.25 \, \mathrm{m} \), and \( h = 0.750 \, \mathrm{m} \). Substitute these into the volume formula: \[ V = \frac{1}{3} \times 0.750 \times (2.50^2 + 2.50 \times 3.25 + 3.25^2) \]

Step 3: Calculate Each Term

Calculate the terms inside the parenthesis: \( 2.50^2 = 6.25 \), \( 3.25^2 = 10.5625 \), and \( 2.50 \times 3.25 = 8.125 \). Add these results: \( 6.25 + 8.125 + 10.5625 = 24.9375 \).

Step 4: Solve for Volume

Substitute the sum back into the formula: \[ V = \frac{1}{3} \times 0.750 \times 24.9375 \] Now calculate: \( \frac{1}{3} \times 0.750 = 0.250 \). Next, multiply: \( 0.250 \times 24.9375 = 6.234375 \).

Step 5: Present the Final Answer

The volume of the frustum of the pyramid, which is the base of the statue, is approximately \( 6.234 \; \mathrm{m}^3 \) after rounding to three decimal places.

Key Concepts

Volume Formula

The volume formula for the frustum of a pyramid is a mathematical gem dating back to ancient Egyptian times. Understanding this formula helps us determine the volume of a frustum, which is a section of a pyramid that lies between two parallel bases. The formula is expressed as: \[ V = \frac{1}{3} h(a^2 + ab + b^2) \] Here, \( V \) represents the volume, \( h \) is the height of the frustum, and \( a \) and \( b \) are the lengths of the sides of the two parallel square bases. This formula essentially calculates the space within the frustum, allowing architects and engineers to determine how much material it might contain or its weight capacity when full. Knowing the volume formula is crucial when dealing with constructions involving frustums, ensuring efficiency and accuracy in design.

Geometry

The study of geometry explores the properties and relations of points, lines, surfaces, and solids. A frustum of a pyramid is a fascinating geometric shape that results when a pyramid is sliced parallel to its base, forming a top base that is smaller and similar to the original base.

• The bases are parallel and maintain proportionate characteristics.
• The height is the perpendicular distance between these two bases.
• The lateral surfaces are trapezoidal in nature.

This shape appears often in architectural designs and sculptures because of its stability and unique aesthetic. To fully grasp the volume calculation, one needs to visualize this three-dimensional shape clearly and understand the layout of its bases and height within the spatial geometry.

Step-by-Step Solution

To find the volume of the frustum of a pyramid, we follow a clear, methodical approach. Let's go through this step-by-step process: 1. **Identify the Given Values:** The given values are \( a = 2.50 \; \mathrm{m} \), \( b = 3.25 \; \mathrm{m} \), and \( h = 0.750 \; \mathrm{m} \). 2. **Substitute into the Formula:** Insert these values into the volume formula: \[ V = \frac{1}{3} \times 0.750 \times (2.50^2 + 2.50 \times 3.25 + 3.25^2) \] 3. **Calculate Inside the Brackets:** Work out each component: - \( 2.50^2 = 6.25 \) - \( 3.25^2 = 10.5625 \) - \( 2.50 \times 3.25 = 8.125 \) 4. **Add the Results:** Combine these to get \( 24.9375 \). 5. **Complete the Multiplication:** Substitute back: \[ V = \frac{1}{3} \times 0.750 \times 24.9375 \] 6. **Calculate Final Outcome:** - First, \( \frac{1}{3} \times 0.750 = 0.250 \) - Then, \( 0.250 \times 24.9375 = 6.234375 \) Thus, the volume of the frustum base of the statue is approximately \( 6.234 \; \mathrm{m}^3 \), rounded to three decimal places.