What Is The Sector Area Created By The Hands Of A Clock With A Radius Of 9 Inches When The Time Is $4:00$?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
What is the Sector Area Created by the Hands of a Clock with a Radius of 9 Inches When the Time is 4:00?
In this article, we will explore the concept of sector area and how it applies to the hands of a clock. We will calculate the sector area created by the hands of a clock with a radius of 9 inches when the time is 4:00. This problem involves geometry and trigonometry, and we will use these concepts to find the solution.
Sector area is a concept in geometry that refers to the area of a sector of a circle. A sector is a region of a circle bounded by two radii and an arc. The area of a sector can be calculated using the formula:
A = (θ/360) × πr^2
where A is the area of the sector, θ is the central angle of the sector in degrees, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.
To calculate the sector area, we need to find the central angle of the sector. The central angle is the angle formed by the two radii of the sector. In this case, we are given that the time is 4:00, which means that the hour hand is at the 4 and the minute hand is at the 12.
The hour hand moves 360 degrees in 12 hours, so it moves 30 degrees per hour. Since the time is 4:00, the hour hand has moved 4 × 30 = 120 degrees from the 12.
The minute hand moves 360 degrees in 60 minutes, so it moves 6 degrees per minute. Since the time is 4:00, the minute hand has moved 0 × 6 = 0 degrees from the 12.
The central angle of the sector is the difference between the angles of the hour hand and the minute hand. Therefore, the central angle is 120 - 0 = 120 degrees.
Now that we have the central angle, we can calculate the sector area using the formula:
A = (θ/360) × πr^2
where A is the area of the sector, θ is the central angle of the sector in degrees, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.
Plugging in the values, we get:
A = (120/360) × π(9)^2 A = (1/3) × π(81) A = (1/3) × 3.14 × 81 A = 84.84
However, this is not one of the answer choices. Let's try to simplify the calculation.
A = (1/3) × π(81) A = (1/3) × 81π 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 problem involves geometry and trigonometry, and we used the formula for sector area to find the solution.
The correct answer is C. 27π in^2.
Frequently Asked Questions (FAQs) About Sector Area and Clock Hands
In our previous article, we explored the concept of sector area and how it applies to the hands of a clock. We calculated the sector area created by the hands of a clock with a radius of 9 inches when the time is 4:00. In this article, we will answer some frequently asked questions (FAQs) about sector area and clock hands.
A: The formula for sector area is:
A = (θ/360) × πr^2
where A is the area of the sector, θ is the central angle of the sector in degrees, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.
A: To calculate the central angle of a sector, you need to find the angle formed by the two radii of the sector. You can do this by subtracting the angle of the minute hand from the angle of the hour hand.
A: The hour hand moves 360 degrees in 12 hours, so it moves 30 degrees per hour. The minute hand moves 360 degrees in 60 minutes, so it moves 6 degrees per minute.
A: To calculate the sector area when the time is 3:00, you need to find the central angle of the sector. The hour hand has moved 3 × 30 = 90 degrees from the 12, and the minute hand has moved 0 × 6 = 0 degrees from the 12. The central angle is 90 - 0 = 90 degrees. Using the formula for sector area, you get:
A = (90/360) × π(9)^2 A = (1/4) × π(81) A = 20.25π
A: To calculate the sector area when the time is 6:00, you need to find the central angle of the sector. The hour hand has moved 6 × 30 = 180 degrees from the 12, and the minute hand has moved 0 × 6 = 0 degrees from the 12. The central angle is 180 - 0 = 180 degrees. Using the formula for sector area, you get:
A = (180/360) × π(9)^2 A = (1/2) × π(81) A = 40.5π
A: Yes, you can use a calculator to calculate the sector area. Simply plug in the values into the formula and solve for the area.
In conclusion, sector area is an important concept in geometry and trigonometry. By understanding the formula for sector area and how to calculate the central angle, you can solve problems involving clock hands. We hope this article has helped you to better understand sector area and clock hands.
- For more information on sector area, visit the website of your local math department or search online for resources.
- To practice calculating sector area, try solving problems on websites such as Khan Academy or Mathway.
- If you have any further questions or need help with a specific problem, feel free to ask a math teacher or tutor.