What Is The Sector Area Created By The Hands Of A Clock With A Radius Of 9 Inches When The Time Is 4 : 00 4:00 4 : 00 ?A. 6.75 Π 6.75 \pi 6.75 Π In. 2 ^2 2 B. 20.25 Π 20.25 \pi 20.25 Π In. 2 ^2 2 C. 27 Π 27 \pi 27 Π In. 2 ^2 2 D. $81

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Understanding the Sector Area of a Clock

When it comes to calculating the area of a sector of a circle, we often rely on the formula A = (θ/360)πr^2, where A is the area of the sector, θ is the central angle in degrees, and r is the radius of the circle. However, in this problem, we are given a specific scenario where the time is 4:00, and we need to find the sector area created by the hands of a clock with a radius of 9 inches. In this article, we will delve into the world of clock geometry and explore how to calculate the sector area in this unique situation.

The Anatomy of a Clock

Before we dive into the calculation, let's take a closer look at the anatomy of a clock. A clock is essentially a circle divided into 12 equal sections, each representing an hour. The hour hand moves 360 degrees in 12 hours, which means it moves 30 degrees per hour. The minute hand, on the other hand, moves 360 degrees in 60 minutes, which means it moves 6 degrees per minute.

Calculating the Central Angle

At 4:00, the hour hand is at the 4 o'clock position, and the minute hand is at the 12 o'clock position. To find the central angle θ, we need to calculate the angle between the hour hand and the minute hand. Since the hour hand moves 30 degrees per hour, it has moved 120 degrees from the 12 o'clock position. The minute hand, being at the 12 o'clock position, has moved 0 degrees. Therefore, the central angle θ is 120 degrees.

Calculating the Sector Area

Now that we have the central angle θ, we can use the formula A = (θ/360)πr^2 to calculate the sector area. Plugging in the values, we get:

A = (120/360)π(9)^2 A = (1/3)π(81) A = 27π

In conclusion, the sector area created by the hands of a clock with a radius of 9 inches when the time is 4:00 is 27π in.^2. This calculation is based on the formula A = (θ/360)πr^2, where θ is the central angle in degrees, and r is the radius of the circle. By understanding the anatomy of a clock and calculating the central angle, we can accurately determine the sector area.

The correct answer is C. 27π in.^2.

This problem requires a good understanding of clock geometry and the ability to apply mathematical formulas to real-world scenarios. The calculation involves finding the central angle and then using the formula to calculate the sector area. This type of problem is essential in mathematics, as it helps students develop problem-solving skills and apply mathematical concepts to everyday life.

  • When dealing with clock geometry, it's essential to understand the movement of the hour and minute hands and how they relate to the central angle.
  • The formula A = (θ/360)πr^2 is a fundamental concept in mathematics, and understanding how to apply it is crucial in solving problems like this.
  • Practice makes perfect, so make sure to practice calculating sector areas with different radii and central angles to become proficient in this area of mathematics.
    Clock Geometry Q&A =====================

Frequently Asked Questions

In this article, we will address some of the most common questions related to clock geometry and the calculation of sector areas.

Q: What is the formula for calculating the sector area of a clock?

A: The formula for calculating the sector area of a clock is A = (θ/360)πr^2, where A is the area of the sector, θ is the central angle in degrees, and r is the radius of the circle.

Q: How do I find the central angle θ in a clock geometry problem?

A: To find the central angle θ, you need to calculate the angle between the hour hand and the minute hand. You can do this by finding the position of the hour hand and the minute hand and then subtracting the smaller angle from the larger angle.

Q: What is the radius of a clock?

A: The radius of a clock is the distance from the center of the clock to the edge of the clock. In most clocks, the radius is 9 inches.

Q: How do I calculate the sector area when the time is 3:00?

A: At 3:00, the hour hand is at the 3 o'clock position, and the minute hand is at the 12 o'clock position. The central angle θ is 90 degrees. Using the formula A = (θ/360)πr^2, we get:

A = (90/360)π(9)^2 A = (1/4)π(81) A = 20.25π

Q: What is the sector area when the time is 6:00?

A: At 6:00, the hour hand is at the 6 o'clock position, and the minute hand is at the 12 o'clock position. The central angle θ is 180 degrees. Using the formula A = (θ/360)πr^2, we get:

A = (180/360)π(9)^2 A = (1/2)π(81) A = 40.5π

Q: Can I use the formula A = (θ/360)πr^2 for any central angle θ?

A: Yes, you can use the formula A = (θ/360)πr^2 for any central angle θ. However, you need to make sure that the central angle θ is in degrees and that the radius r is in the same units as the area A.

Q: What is the sector area when the time is 12:00?

A: At 12:00, the hour hand and the minute hand are at the 12 o'clock position. The central angle θ is 0 degrees. Using the formula A = (θ/360)πr^2, we get:

A = (0/360)π(9)^2 A = 0

In this article, we have addressed some of the most common questions related to clock geometry and the calculation of sector areas. We have provided formulas and examples to help you understand how to calculate the sector area of a clock. Remember to practice and become proficient in using the formula A = (θ/360)πr^2 to solve problems like this.

  • Make sure to understand the movement of the hour and minute hands and how they relate to the central angle.
  • Practice calculating sector areas with different radii and central angles to become proficient in this area of mathematics.
  • Use the formula A = (θ/360)πr^2 to calculate the sector area of a clock, and make sure to use the correct units for the central angle θ and the radius r.