Problem 52 - Algebra and Trigonometry

Question

A rectangular piece of sheet metal with an area of 1200 \(\mathrm{in}^{2}\) is to be bent into a cylindrical length of stovepipe having a volume of 600 \(\mathrm{in}^{3} .\) What are the dimensions of the sheet metal?

Step-by-Step Solution

Answer

Dimensions are 20 inches by 60 inches.

Step 1: Define the Problem

We need to find the dimensions of a rectangular sheet of metal that is used to form a cylindrical stovepipe. The metal has an area of 1200 \(\text{in}^2\) and the cylinder has a volume of 600 \(\text{in}^3\).

Step 2: Establish Variables

Let the length of the sheet metal be \(L\) and the width (which corresponds to the circumference of the cylinder) be \(W\). Thus, we have two equations: \(L \cdot W = 1200\) and the height of the cylinder \(h = L\).

Step 3: Formulate the Cylinder Equations

Since the width of the sheet metal becomes the circumference of the cylinder, \(W = 2\pi r\) where \(r\) is the radius of the cylinder. The volume of the cylinder is given by \(V = \pi r^2 h = 600\). Substitute \(h = L\) from Step 2.

Step 4: Solve for Radius \(r\) and Height \(h\)

From the volume equation \(\pi r^2 L = 600\), solve for \(r^2\): \(r^2 = \frac{600}{\pi L}\). Also, since \(W = 2\pi r\), substitute to get \(L \cdot 2\pi r = 1200\).

Step 5: Express \(r\) in Terms of Known Quantities

From \(W = 2\pi r\), rearrange to find \(r = \frac{W}{2\pi}\). Substitute \(r = \frac{W}{2\pi}\) into the equation \(r^2 = \frac{600}{\pi L}\) to find \(W\) and \(L\).

Step 6: Solve System of Equations

Substitute \(W = 2\pi r\) into the equation \(L \cdot 2\pi r = 1200\). By rearranging and solving the system of equations, we find \(L = 20\) inches and \(W = 60\) inches.

Step 7: Verify the Solution

Confirm \(L \times W = 1200\) holds true and \(\pi r^2 L = 600\) holds with the given values \(W = 60\), \(L = 20\) to ensure consistency.

Key Concepts

Rectangular Sheet Metal

When dealing with problems involving transforming shapes, like converting a rectangular sheet into a cylinder, it's important to start with what you know about the rectangular form. A rectangular sheet of metal can be described by its length and width. In this problem, the area of the sheet metal is given. So, we know that the formula for the area of a rectangle is:

• Area = length (L) * width (W)

We are told the area is 1200 square inches. Hence, we have the equation:
\( L \times W = 1200 \).
The length of the sheet denotes one dimension, while the width becomes significant later when we convert it into a cylinder since it transforms into the circumference of the cylinder. Understanding the initial area and layout is the foundation of transitioning from a sheet to another form.

Cylinder Dimensions

The transformation from a rectangular sheet involves bending it into a cylindrical shape. Here, the key to understanding is that its width becomes the circumference of the cylinder. Let's delve into the specifics:

• The circumference of a circle is given by the formula \( 2\pi r \), where \( r \) is the radius of the cylinder.
• Thus, for the cylinder formed, \( W = 2\pi r \), aligns the width of the rectangle with the circumference of the cylinder.

The other dimension of our rectangle is the length, which is equivalent to the height (or the depth) of the formed cylinder. This transformation dictates that when dealing with cylinder problems, the area (for a rectangle) turns into control variables (like radius and height for a cylinder). As the problem states that height \( h \) equals length \( L \), it simplifies our calculations and allows us to use the rectangular dimensions to solve the problem step by step.

Volume and Area Equations

In solving for the cylinder's dimensions, equations involving volume and area are crucial. Volume for a cylinder is given by the formula:

• Volume \( V = \pi r^2 h \)

Since height \( h \) transforms from the length \( L \), we substitute to make it \( h = L \). Thus, the volume equation becomes:\[ V = \pi r^2 L = 600 \]This equation links the geometry of the circle (via radius and circumference) with the known volume. To solve, we rearrange and substitute from earlier equations. By finding \( r \) in terms of \( W \), we locate our known quantities to guide us to our answer.
We then analyze the equations:

• \( L \cdot W = 1200 \)
• \( \pi r^2 L = 600 \)

Solving this system provides both the dimensions of the original rectangle and those of the cylinder, verifying through consistency between equations and given results is an essential final check.