Problem 42 - Basic Technical Mathematics with Calculus

Question

\(\text {Solve the given problems.}\) What is the area of a paper label that is to cover the lateral surface of a cylindrical can 3.00 in. in diameter and 4.25 in. high? The ends of the label will overlap 0.25 in. when the label is placed on the can.

Step-by-Step Solution

Answer

The area of the paper label is approximately 41.10 square inches.

Step 1: Calculate the Circumference of the Cylinder

To find the lateral surface area, we first need the circumference of the base of the cylinder. The formula for circumference is \( C = \pi d \), where \( d \) is the diameter. Here, \( d = 3.00 \) in. So, \( C = \pi \times 3.00 = 3\pi \) inches.

Step 2: Adjust for the Overlap in the Paper Label

The label overlaps by 0.25 inches. Therefore, the effective length of the paper label that is required to cover around the cylinder will be longer than the cylinder's circumference. The length needed is \( C + 0.25 = 3\pi + 0.25 \) inches.

Step 3: Calculate the Lateral Surface Area of the Cylindrical Label

The lateral surface area of the cylindrical label is calculated by multiplying the circumference (adjusted for overlap) by the height of the cylinder. Thus, the lateral surface area \( A \) is given by \( A = \text{effective length} \times h = (3\pi + 0.25) \times 4.25 \).

Step 4: Substitute the Values and Solve

Substitute \( \pi \approx 3.14 \) into the lateral surface area formula: \[ A = (3 \times 3.14 + 0.25) \times 4.25 = (9.42 + 0.25) \times 4.25 = 9.67 \times 4.25. \] Calculate to find \( A \approx 41.0975 \) square inches.

Key Concepts

Circumference Calculation

Understanding how to calculate the circumference is crucial in solving problems about areas involving cylinders. The circumference is the distance around the circular base of a cylinder. Its formula is quite straightforward, which is: \[ C = \pi d \] where \( d \) is the diameter of the circle. In many real-world problems, you'll often be given the diameter as straightforward like in this case, 3.00 inches. To get the circumference, you simply multiply this diameter by \( \pi \). Remember that \( \pi \) is an irrational number approximately equal to 3.14159, but in homework, you can use 3.14 for easier calculations. Thus, here the calculation goes:

• Diameter \( d = 3.00 \) inches
• Circumference \( C = \pi \times 3.00 = 3\pi \)

So, for our exercise, the circumference of the can's base is 3 times \( \pi \), which forms the foundation for further calculations.

Surface Area Formula

The lateral surface area of a cylinder is the area of the rectangle that wraps around the circular base. To find the lateral surface area, you multiply the circumference of the base by the height of the cylinder. Here's the formula you'll need:

• Lateral Surface Area \( A = C \times h \)

where \( h \) is the height. For the cylinder in our problem:

• Height \( h = 4.25 \) inches
• Circumference (adjusted for overlap, we'll cover this in detail in the next section)

The goal is to cover the curved surface of the cylinder. The solution required this lateral area for the label that smoothly fits around the can's surface. This understanding is vital, especially in labeling processes in industries.

Overlap Adjustment

Overlaps are common in labeling to ensure the label stays secure around the cylindrical object. In our problem, there is an overlap of 0.25 inches. This means when determining how much label material you need, you add the overlap distance to the circumference, giving you the effective length of the label. By doing this, we prevent any gaps that could occur if the label were only as long as the circumference. So, the math looks like this:

• Original Circumference \( C = 3\pi \)
• Overlap = 0.25 inches

When you add the overlap, the new effective length becomes:

• Effective length = \( C + 0.25 \)

So, in the context of solving for the label's surface area, this step ensures you're not underestimating the material needed. This adjustment is quintessential for accuracy in practical and real-life scenarios, making sure your labels fit snugly and effortlessly.