Problem 33 - Basic Technical Mathematics with Calculus

Question

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. A highway marking pylon has the shape of a frustum of a cone. Find its centroid if the radii of its bases are \(5.00 \mathrm{cm}\) and \(20.0 \mathrm{cm}\) and the height between bases is \(60.0 \mathrm{cm}\).

Step-by-Step Solution

Answer

The centroid is located 17.86 cm from the smaller base, along the frustum's height.

Step 1: Understand the Problem

We need to find the centroid of a frustum of a cone, which is essentially a cone with its top cut off. The frustum has two circular bases with known radii and a given height between these bases.

Step 2: Identify Formula for Centroid of a Frustum

The formula to find the centroid \(C_z\) of a frustum of a cone along its height (z-axis) is given by:\[C_z = \frac{h}{4} \left( \frac{r^2 + 2rR + R^2}{r^2 + rR + R^2} \right)\]where \(h\) is the height of the frustum, \(r\) is the radius of the smaller base, and \(R\) is the radius of the larger base.

Step 3: Input Known Values

For the given frustum, we have:\(r = 5.00 \, \text{cm}\), \(R = 20.0 \, \text{cm}\), and \(h = 60.0 \, \text{cm}\).Substitute these values into the formula.

Step 4: Calculate the Numerator of the Formula

Calculate the numerator value in the formula:\(r^2 + 2rR + R^2 = 5^2 + 2(5)(20) + 20^2 = 25 + 200 + 400 = 625\).

Step 5: Calculate the Denominator of the Formula

Calculate the denominator value in the formula:\(r^2 + rR + R^2 = 5^2 + (5)(20) + 20^2 = 25 + 100 + 400 = 525\).

Step 6: Plug the Values into the Formula

Substitute the calculated numerator and denominator back into the centroid formula:\[C_z = \frac{60}{4} \left( \frac{625}{525} \right)\]Simplifying this gives:\(C_z = 15 \times \frac{625}{525} = 17.8571 \, \text{cm}\).

Step 7: Conclusion

The centroid of the frustum is at 17.86 cm from the smaller base along the central vertical axis of the frustum.

Key Concepts

Frustum of a Cone

A frustum of a cone is a three-dimensional geometric shape formed when a smaller, top portion of a cone is removed. This leaves a shape that has two parallel circular ends, the smaller of which we call the smaller base and the larger the larger base.
It is important to note that a frustum retains much of the general symmetry of a cone, which makes it a common choice in structures and everyday items, like highway pylons. Understanding this shape is crucial when calculating attributes such as volume, surface area, or the location of the centroid.

• A frustum has two main measurements:
  • The radii of the two bases
  • The perpendicular height between the two bases
• The bases are parallel and circular.
• The lined edges of the frustum taper from the larger base to the smaller base.

Mastering the properties and formulas related to the frustum of a cone can enhance your geometry skills and deepen your understanding of various applications in mathematics and real life.

Geometry

Geometry is the branch of mathematics that deals with the properties and relationships of points, lines, surfaces, and solids. Studying geometry helps us understand and compute volumes, areas, and the properties of different shapes, like the frustum of a cone.
When analyzing a frustum, geometry is employed to define and understand:

• The circular bases, which require knowledge of circles and their radii and diameters.
• The height, which is a linear measurement connecting the bases perpendicularly.

For many geometric problems, finding the centroid is a common task. The centroid is the geometric center or "balance point" of an object. In frustums and other geometric shapes, the centroid can be located by applying specific mathematical formulas, which simplifies the analysis of complex shapes.
The importance of geometry lies in its vast applications, from architectural designs to everyday item shapes, making it an essential tool in mathematics and science.

Mathematics Problem Solving

Mathematics problem solving is a methodical process requiring you to analyze, think critically, and apply appropriate formulas and concepts to find solutions. When dealing with exercises like calculating the centroid of a frustum of a cone, follow these steps:

• Understand the problem to determine what is being asked and identify given information.
• Select the appropriate formula or method to solve for the unknowns. Here, the centroid formula for a frustum is used.
• Input the given values carefully within the chosen formula and simplify each step methodically.
• Check calculations for accuracy and logical consistency.

This process encourages logical thinking and improves your ability to tackle similar and more complex problems efficiently. Understanding the principles of mathematics problem solving ensures you can approach any new problem with confidence and preparedness.

Formula Application

Applying formulas effectively is crucial in solving mathematical problems like finding the centroid of a frustum of a cone.

• Start by identifying the right formula. For centroids of frustums, use:
  \[C_z = \frac{h}{4} \left( \frac{r^2 + 2rR + R^2}{r^2 + rR + R^2} \right)\]
• Insert the correct values for each variable in the formula. In this case:
  • \(r = 5.00 \, \text{cm}\)
  • \(R = 20.0 \, \text{cm}\)
  • \(h = 60.0 \, \text{cm}\)
• Simplify the fraction by calculating both the numerator and the denominator separately, ensuring accuracy at each stage.
• Perform a final calculation by multiplying and dividing as required.

Correct application of formulas not only leads to accurate answers but also strengthens your understanding of the mathematical processes involved, setting a solid foundation for problem solving in various contexts.