Problem 12 - Basic Engineering Mathematics

Question

A rivet consists of a cylindrical head, of diameter \(1 \mathrm{~cm}\) and depth \(2 \mathrm{~mm}\), and a shaft of diameter \(2 \mathrm{~mm}\) and length \(1.5 \mathrm{~cm}\). Determine the volume of metal in 2000 such rivets.

Step-by-Step Solution

Answer

The volume of metal in 2000 rivets is \(130\pi \text{ cm}^3\).

Step 1: Volume of the Cylindrical Head

The cylindrical head has a diameter of \(1 \text{ cm}\) and a depth of \(2 \text{ mm}\), which is \(0.2 \text{ cm}\). The radius \(r\) is half of the diameter: \(r = \frac{1}{2} = 0.5 \text{ cm}\). The volume \(V_h\) of a cylinder is given by \(V_h = \pi r^2 h\). Substituting the values, we get:\[ V_h = \pi (0.5)^2 (0.2) = \pi \times 0.25 \times 0.2 = 0.05\pi \text{ cm}^3 \].

Step 2: Volume of the Cylindrical Shaft

The shaft has a diameter of \(2 \text{ mm}\), which is \(0.2 \text{ cm}\), and a length of \(1.5 \text{ cm}\). The radius \(r_s\) is \(0.1 \text{ cm}\). The volume \(V_s\) of the shaft is given by \(V_s = \pi r_s^2 h_s\). Substituting we get:\[ V_s = \pi (0.1)^2 (1.5) = \pi \times 0.01 \times 1.5 = 0.015\pi \text{ cm}^3 \].

Step 3: Total Volume of One Rivet

The total volume of one rivet is the sum of the volume of the head and the shaft. Thus, the total volume \(V_t\) is \(V_t = V_h + V_s\). Substituting the values from the previous steps, we have:\[ V_t = 0.05\pi + 0.015\pi = 0.065\pi \text{ cm}^3 \].

Step 4: Volume of 2000 Rivets

To find the volume of 2000 rivets, multiply the volume of one rivet by 2000. Thus, the volume \(V_{2000}\) is:\[ V_{2000} = 2000 \times 0.065\pi = 130\pi \text{ cm}^3 \].

Key Concepts

Cylindrical Shapes

Cylindrical shapes are common in everyday objects and have importance in various applications due to their uniform structure. A cylinder consists of two parallel circular bases and a curved surface connecting the bases. To determine the volume of a cylinder, we utilize the formula \( V = \pi r^2 h \), where \( r \) is the radius of the base and \( h \) is the height or depth of the cylinder.
For example, consider the rivet head as a cylinder. Its diameter is given as \( 1 \text{ cm} \) which translates to a radius of \( 0.5 \text{ cm} \). The depth or height is \( 0.2 \text{ cm} \). By substituting these into the volume formula, we find the volume of just the head.

• Understanding the cylinder as a combination of circular bases and a height helps in visualizing the shape.
• Recognizing that both head and shaft of a rivet can be described as separate cylinders is key in such calculations.

Grasping how these shapes work simplifies the process of calculating their volumes.

Measurement Units Conversion

Converting measurement units is crucial in ensuring accuracy in geometrical calculations. This is especially important when dimensions are given in different units, such as millimeters and centimeters.
The exercise involves measurements in both millimeters and centimeters. We need to convert millimeters to centimeters to maintain consistent units. Knowing that \( 1 \text{ mm} = 0.1 \text{ cm} \), converting is simple: just multiply the millimeter measurement by \( 0.1 \) to get centimeters.
For instance:

• The shaft's diameter of \( 2 \text{ mm} \) gets converted to \( 0.2 \text{ cm} \).
• The depth of the head, originally \( 2 \text{ mm} \), becomes \( 0.2 \text{ cm} \) after conversion.

Accurate conversion ensures calculations are correct, regardless of initial measurement disparities.

Geometry Applications

Geometry applications are prevalent in real-world scenarios such as manufacturing, architecture, and design. In this exercise, understanding geometry enables us to calculate the volume of the metal in the rivets.
The task involved finding the total volume of 2000 rivets by first calculating the volume of a single rivet. The application of formulas and logical steps in geometry aids these complex calculations:

• Calculate the volume of each component (head and shaft) separately using the cylinder volume formula.
• Sum the volumes to find the total volume of one rivet.
• Multiply by the number of rivets to find the total volume.

This systematic approach reveals how individual component calculations contribute to larger scale productions in manufacturing.