Problem 46 - Precalculus Mathematics for Calculus

Question

A rectangular piece of sheet metal with an area of 1200 in \(^{2}\) is to be bent into a cylindrical length of stovepipe having a volume of 600 in \(^{3} .\) What are the dimensions of the sheet metal? (IMAGE CAN'T COPY).

Step-by-Step Solution

Answer

The dimensions of the sheet are approximately 12.57 in (width) and 75 in (length).

Step 1: Understand the problem

We have a rectangular sheet of metal with an area of 1200 in². This sheet is bent to form a cylindrical pipe with a volume of 600 in³. We need to determine the dimensions (length and width) of this sheet.

Step 2: Express the area formula

The area of the rectangular sheet, which will be the lateral surface area of the cylinder, is given by the formula: \ A = l \times w \, where \( l \) is the length and \( w \) is the width. Given that \( A = 1200 \text{ in}^2 \), we have: \ l \times w = 1200.

Step 3: Relate the cylinder's dimensions

When the sheet metal is bent into a cylinder, one dimension will be the circumference of the base \( C \), and the other will be the height \( h \) of the cylinder. So, \( C = 2\pi r \), where \( r \) is the radius of the cylinder.

Step 4: Use the volume formula for a cylinder

The volume \( V \) of the cylinder is given by: \ V = \pi r^2 h \ Using \( V = 600 \), we have \( \pi r^2 h = 600 \).

Step 5: Substitute circumference into area equation

The circumference of the cylinder (width of the rectangular sheet) is \( w = 2\pi r \). Substitute into the area equation: \ l \times 2\pi r = 1200. \ Rearrange to find \( l \) : \ l = \frac{1200}{2\pi r} = \frac{600}{\pi r}.

Step 6: Combine with volume expression

From \( \pi r^2 h = 600 \) and knowing \( h = l = \frac{600}{\pi r} \), substitute into the volume equation: \ \pi r^2 \left(\frac{600}{\pi r}\right) = 600. \ Simplify to get \( r = 2 \).

Step 7: Calculate remaining dimensions

Substitute \( r = 2 \) into \( w = 2\pi r \) to find \( w = 4\pi \) and into \( h = \frac{600}{\pi r} \) to find \( h = 75 \). Therefore, the dimensions of the sheet are width \( 2\pi r = 4\pi \) and length \( 75 \).

Key Concepts

Rectangular Sheet Metal

When you think about a rectangular sheet of metal, you might imagine a flat piece, like a large fan blade but without the curves. This sheet of metal has certain attributes that are very important in cylindrical calculations, such as area. We often start by considering the area, as it tells us how much space the metal covers when flat.

• The area of a rectangle is calculated as the product of its length and width, expressed by the equation: \[ A = l \times w \]
• In the exercise, the sheet has an area of 1200 square inches. This means when you multiply the length ( \( l \)) by the width ( \( w \)), you should get 1200.

Understanding these basics is crucial, as this area becomes the lateral surface part of the cylinder once the metal is bent into its cylindrical shape.

Cylinder Volume

Calculating the volume of a cylinder involves understanding the space inside the cylinder. It's akin to knowing how much liquid a soup can can hold, considering its height and the radius of its circular base.

• The volume ( \( V \)) of a cylinder is given by the formula: \[ V = \pi r^2 h \] Here, \( r \) stands for the radius of the base, and \( h \) stands for height.
• In the problem, the cylinder must have a volume of 600 cubic inches, which tells us the exact space the formed cylinder should enclose.

To find the dimensions of our metal that achieve this volume, we incorporated this formula into our problem-solving process.

Cylinder Surface Area

When the sheet metal is bent, its surface area doesn't disappear – it transforms into the cylinder's lateral surface. This surface essentially wraps around the cylinder, forming its outer shell.

• The lateral surface area of a cylinder, derived from the metal sheet, can also be expressed in terms of its dimensions.
• Since the lateral surface is made from the original rectangular sheet with an area of 1200 square inches, this remains constant and forms both the height and circumference of the cylinder.

By using this relationship, we can solve for different parts of the cylinder's geometry, such as height or circumference, based on the given area.

Circumference and Radius Relationship

In a cylindrical object, the circumference of its base is a critical relationship in linking its dimensions. This is especially important when dealing with transformations from a flat sheet to a curved object, like a cylinder.

• The circumference ( \( C \)) of a cylinder's base circle is calculated using the radius ( \( r \)), where \[ C = 2\pi r \]
• In the case of our rectangular sheet, once bent, the width of the sheet becomes the circumference of this cylinder.
• This implies that the width \( w \) equates to the circumference \( 2\pi r \) of the cylinder.

Understanding this relationship helps us connect the linear measurements of the sheet with the more complex round structure of the cylinder, smoothing the path to calculating vital dimensions like radius and width.