Problem 21 - Basic Engineering Mathematics

Question

The volume \(V \mathrm{~cm}^{3}\) of a right circular cone is given by \(V=\frac{1}{3} \pi r^{2} h\). Given that \(r=4.321 \mathrm{~cm}\) and \(h=\) \(18.35 \mathrm{~cm}\), find the volume, correct to 4 significant figures.

Step-by-Step Solution

Answer

The volume is 358.8 cm³.

Step 1: Substitute Values into Volume Formula

Start with the volume formula for a cone: \( V = \frac{1}{3} \pi r^2 h \). Substitute the given values for \( r = 4.321 \) cm and \( h = 18.35 \) cm.

Step 2: Calculate the Square of the Radius

Calculate \( r^2 \) using \( r = 4.321 \) cm: \( r^2 = (4.321)^2 = 18.668641 \) cm².

Step 3: Multiply by Height

Multiply the square of the radius by the height: \( 18.668641 \times 18.35 = 342.69584635 \) cm³.

Step 4: Calculate Volume Before Rounding

Apply the formula \( V = \frac{1}{3} \pi \times 342.69584635 = \frac{1}{3} \times 3.141592653589793 \times 342.69584635 \). This results in \( V \approx 358.7710987 \) cm³ before rounding.

Step 5: Round Result to Significant Figures

Round the calculated volume, \( 358.7710987 \), to 4 significant figures to get \( 358.8 \) cm³.

Key Concepts

Understanding Geometry in Volume Calculation

Geometry is all about shapes and sizes. In the context of this exercise, we are focusing on a cone. A cone is a three-dimensional (3D) geometric shape with a circular base connected by a curved surface to a single vertex, like an ice cream cone. This shape is significant because its volume relies on the size of its base and height.
In our problem, we have a right circular cone, meaning the tip (vertex) of the cone is directly above the center of its circular base. This orientation simplifies our calculations, both in theory and in practice. By understanding these properties, we can intuitively grasp the steps and why the formula is structured as it is.

Exploring the Mathematical Formula for a Cone's Volume

The volume \( V \) of a cone is given by the formula \( V = \frac{1}{3} \pi r^2 h \), where:

• \( r \) is the radius of the base of the cone.
• \( h \) is the height of the cone.

This formula arises from the volume of a cylinder, \( V = \pi r^2 h \), noting that a cone is essentially one-third of a cylinder with the same base and height. The factor \( \frac{1}{3} \) is key here to reflect this relationship.
Thus, to find the volume, we first calculate the area of the base (\( \pi r^2 \)), and then use this area to find the space occupied by the cone's height, adjusting with the one-third factor.

Significance of Significant Figures

Significant figures have a central role in scientific calculations, as they indicate the precision of a measurement or calculation. When asked to give an answer to a specific number of significant figures, it ensures that our result aligns with the precision of the provided data.
In our exercise, the volume needs to be rounded to four significant figures. This means that we should express the volume while maintaining four digits that are reliable and meaningful, starting with the first non-zero digit. For example, after calculation, we achieved a volume of approximately 358.7710987 cubic centimeters, which should be rounded to 358.8 cubic centimeters when presented to four significant figures.

Step-by-Step Calculation Process

Performing calculations step-by-step is vital for accuracy, especially when dealing with geometric problems. Let's revisit the steps:

• Step 1: Substitute Known Values - The formula \( V = \frac{1}{3} \pi r^2 h \) requires the values of \( r \) and \( h \). For our problem, \( r = 4.321 \) cm and \( h = 18.35 \) cm.
• Step 2: Calculate \( r^2 \) - Multiply the radius by itself: \( r^2 = (4.321)^2 = 18.668641 \) cm².
• Step 3: Multiply by Height - Multiply the area \( r^2 \) by the height \( h \): \( 18.668641 \times 18.35 = 342.69584635 \) cm³.
• Step 4: Determine Volume Prior to Rounding - Calculate \( V = \frac{1}{3} \pi \times 342.69584635 \), which results in approximately 358.7710987 cm³.
• Step 5: Round to Significant Figures - Adjust the volume to four significant figures, providing the final result of 358.8 cm³.

This systematic approach ensures clarity and reduces error, particularly in complex calculations.