Problem 20 - Basic Engineering Mathematics

Question

A lampshade is in the shape of a frustum, of a cone. The vertical height of the shade is \(25.0 \mathrm{~cm}\) and the diameters of the ends are \(20.0 \mathrm{~cm}\) and \(10.0 \mathrm{~cm}\), respectively. Determine the area of the material needed to form the lampshade, correct to 3 significant figures.

Step-by-Step Solution

Answer

The area needed is approximately 1199 cm^2.

Step 1: Calculate the Radii

Let's identify the radii of the top and bottom circles of the frustum. The radii are half of the given diameters. For the top circle, the diameter is \(20.0\, \text{cm}\), so the radius \(r_1 = \frac{20.0}{2} = 10.0\, \text{cm}\). For the bottom circle, the diameter is \(10.0\, \text{cm}\), giving \(r_2 = \frac{10.0}{2} = 5.0\, \text{cm}\).

Step 2: Calculate the Slant Height

To find the lateral surface area of a frustum, we need the slant height \(l\). We use the Pythagorean theorem in the triangle formed by the height, the difference in radii, and the slant height: \(l = \sqrt{(r_1 - r_2)^2 + h^2}\). Here \(h = 25.0\, \text{cm}\), so \(l = \sqrt{(10.0 - 5.0)^2 + 25.0^2} = \sqrt{25 + 625} = \sqrt{650}\, \text{cm}.\) Thus, \(l \approx 25.495\, \text{cm}.\)

Step 3: Calculate the Lateral Surface Area

The lateral surface area \(A\) of the frustum is given by \(\pi (r_1 + r_2) l\). Plug in the values: \(A = \pi (10.0 + 5.0) \times 25.495 = 15\pi \times 25.495\). Calculate \(A \approx 15 \times \pi \times 25.495 \approx 1200\, \text{cm}^2\).

Step 4: Round to Three Significant Figures

Round the calculated lateral surface area to three significant figures: The rounded value is \(A \approx 1199\, \text{cm}^2\).

Key Concepts

Lateral Surface Area

When dealing with a frustum of a cone, the lateral surface area is a critical concept to understand, especially when you're trying to determine the material needed for a physical object like a lampshade. The lateral surface area is essentially the "side" or "curved" surface of the frustum without the top and bottom bases.
To compute this area, you use the formula:

• \[A = \pi(r_1 + r_2)l\]

where \(r_1\) and \(r_2\) are the radii of the top and bottom circular ends, and \(l\) is the slant height. The term \(\pi(r_1 + r_2)\) resembles the perimeter for the lateral circular section, and multiplying it by the slant height gives you the total surface area. In our lampshade example, after calculating the radii and slant height, we can plug those values into the formula to get the lateral surface area.

Slant Height Calculation

The slant height of a frustum is a real essential measurement that frequently puzzles students. It's the distance measured along the side of the frustum and is needed to find the lateral surface area. To find the slant height in a frustum, consider the sectional view of the frustum forming an inclined triangle.

• The base of this triangle is the width between the radii of the top and bottom circles, \(r_1 - r_2\).
• The vertical or height component is the given height of the frustum, denoted as \(h\).

Applying the Pythagorean theorem to this triangle, the slant height \(l\) can be calculated with:

• \[l = \sqrt{(r_1 - r_2)^2 + h^2}\]

In the provided lampshade example, substituting the given values helps in computing \(l\) to be approximately 25.495 cm, allowing you to progress with finding the surface area.

Pythagorean Theorem

The Pythagorean theorem is an incredibly useful tool in geometry, applicable in various scenarios, including the calculations involving the frustum.**Understanding the Theorem**:The theorem is articulated mathematically as:

• \[c^2 = a^2 + b^2\]

where \(c\) is the hypotenuse of a right triangle, and \(a\) and \(b\) are the other two sides. It's a reliable method for us to find lengths when we have right-angled triangles.
In the context of a frustum, the Pythagorean theorem helps calculate the slant height, bridging the gap between the height of the frustum and the difference in the radii (which leads to forming a right triangle when visualized).
By substituting the values into the theorem, one can find the slant height necessary for further calculations like the lateral surface area, demonstrating how fundamental geometric principles simplify the task.