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Introduction

In this tutorial, we will explore how to determine the length of the rope that wraps around two wheels. This problem relates to the concept of tangent lines and circles, an essential topic in geometry for Grade 8 mathematics. Understanding how to apply these concepts is not only crucial for solving similar problems but also for honing your mathematical skills.

Step 1: Understand the Problem

Before diving into calculations, clarify the given information:

• There are two wheels, each with a radius of 21 cm.
• A rope is wrapped around both wheels.

Key Considerations

• The rope's length corresponds to the distance around the wheels.
• The wheels are tangential to each other, meaning they touch at one point without overlapping.

Step 2: Calculate the Circumference of One Wheel

The circumference of a circle can be calculated using the formula: [ C = 2 \pi r ] Where:

• ( C ) is the circumference
• ( r ) is the radius
• ( \pi ) is approximately 3.14

Calculation

1. For one wheel with a radius of 21 cm: [ C = 2 \times \pi \times 21 = 2 \times 3.14 \times 21 ] [ C \approx 131.88 \text{ cm} ]

Step 3: Calculate the Total Rope Length

Since there are two wheels, the total length of the rope will be the sum of the circumferences of both wheels.

Calculation

1. Total length of the rope: [ \text{Total Length} = C_1 + C_2 = 131.88 + 131.88 ] [ \text{Total Length} \approx 263.76 \text{ cm} ]

Step 4: Consider the Tangential Section

Since the wheels are tangential, we also need to account for the straight section of the rope between the two wheels.

Calculation

1. The distance between the centers of the two wheels is equal to the sum of their radii: [ \text{Distance} = 21 \text{ cm} + 21 \text{ cm} = 42 \text{ cm} ]

2. The total length of the rope becomes: [ \text{Total Length} = C_1 + C_2 + \text{Distance} = 263.76 \text{ cm} + 42 \text{ cm} ] [ \text{Total Length} \approx 305.76 \text{ cm} ]

Conclusion

In summary, the total length of the rope that wraps around the two wheels is approximately 305.76 cm. Understanding the relationship between the circumference of circles and the geometry of tangents is vital for solving these types of problems. You can further practice similar problems to strengthen your understanding of geometry concepts.