Sylvia conducts an experiment using two long pieces of a cord and a spherical ball that has been cut in half. The radius of the spherical ball is 8 centimeters. She wraps the first cord around the hemisphere of the ball until it is completely covered, as shown in Figure 1. She cuts the cord and removes it from the ball. She then wraps the second cord around the bottom circle of the ball until it is completely covered, as shown in Figure 2. She again cuts the cord and removes it from the circular base. Which of the following correctly compares the lengths of the cords and best justifies the reason for this relationship? A - The length of the first cord must be approximately equal to the length of the second cord because the surface area of the hemisphere is the same as the area of its circular base. B - The length of the first cord must be approximately 2 times the length of the second cord because the surface area of the hemisphere is 2 times the area of its circular base. C - The length of the first cord must be approximately 4 times the length of the second cord because the surface area of the hemisphere is 4 times the area of its circular base. D - The length of the first cord must be approximately 8 times the length of the second cord because the surface area of the hemisphere is 8 times the area of its circular base.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.2: Ellipses

Problem 66E

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Sylvia conducts an experiment using two long pieces of a cord and a spherical ball that has been cut in half. The radius of the spherical ball is 8 centimeters. She wraps the first cord around the hemisphere of the ball until it is completely covered, as shown in Figure 1. She cuts the cord and removes it from the ball. She then wraps the second cord around the bottom circle of the ball until it is completely covered, as shown in Figure 2. She again cuts the cord and removes it from the circular base.

Which of the following correctly compares the lengths of the cords and best justifies the reason for this relationship?

A - The length of the first cord must be approximately equal to the length of the second cord because the surface area of the hemisphere is the same as the area of its circular base.

B - The length of the first cord must be approximately 2 times the length of the second cord because the surface area of the hemisphere is 2 times the area of its circular base.

C - The length of the first cord must be approximately 4 times the length of the second cord because the surface area of the hemisphere is 4 times the area of its circular base.

D - The length of the first cord must be approximately 8 times the length of the second cord because the surface area of the hemisphere is 8 times the area of its circular base.

### Understanding Coiled Structures

In the figures presented, we observe two different coiling methods of a strip.

#### Figure 1: Conical Coiling
Figure 1 illustrates a conical coiling method where the strip is wound in a conical shape, forming several layers that ascend in a stacked manner resembling a beehive structure. This kind of coiling is used in various applications, such as in the construction of traditional pottery or rope winding.

#### Figure 2: Cylindrical Coiling
In contrast, Figure 2 shows a cylindrical coiling technique, where the strip is coiled flatly around a central point in a more compact manner, forming concentric circles. This type of coiling is common in tapes, hoses, and other flexible materials that need to be stored efficiently.

Both diagrams serve to visually represent different ways in which flexible materials can be coiled for storage or structural purposes. Understanding these methods is essential in fields such as material science, engineering, and even in everyday problem-solving situations.

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